Christian Teachings about Judaism

How can I find the Christian teachings about Judaism in the biblical text using Logos?

I can't make sense of this question because the consensus is that from the 1st-4th centuries Christianity was diverging from Judaism and becoming a separate religion. Therefore, during the period that the New Testament was written, everything written by Christians would be viewed as Jewish of a Christian sect.

Anyone can prooftext the Bible enough to get a set ruling of understanding about Judaism.

A misunderstanding you mean. I find it best to think of three Judaisms: 

1. The religion of the early Israelites
2. The second Temple religion
3. Rabbinic Judaism with "sidebars" of Karaite Judaism and Beta Israel

Judaism had to make a radical shift to become a religion not based upon a sacrificial system. To know anything about Judaism as it has existed contemporaneously with Christianity, one needs to be familiar with the canonical rabbinic literature. Rabbinic Judaism, the Judaism we know, evolved in exile after the destruction of the Temple i.e. outside the time period of the writing of the New Testament and therefore not discussed in the Christian Bible.

But Jews do not read the same texts as Christians do

Many of the texts are the same; the assumptions and methods are not. And the view of Oral Torah, which varies within Christianity, is a major unknown in this statement.

I know of several Jewish scholars who reject the idea that Christianity was once a Jewish sect ... they treat it more like a cult of personality much as we think of the Branch Davidians.

As for early Christian understanding of Judaism, I would look at:

• Justin Martyr - "Dialogue with Trypho"
• Irenaeus of Lyon - "Against Heresies"
• Tertullian "Against Marcion"
• Eusebius of Caesarea - "Ecclesiastical History"

For Christians emphasizing the role of Judaism in Christianity, I would look at:

• Clement of Alexandria - "Stromata" with Greek philosophy serving an intermediary role
• Origen of Alexandria - often tries to seek consensus with Jewish scholars
• Jerome - at times relied on Jewish interpretation to inform his translation
• Augustine of Hippo

Bonus for Jewish identification of "insurmountable" differences:

• Samson Raphael Hirsch - sees Christians as rejecting monotheism (Trinity) and Oral Torah
• Rabbi Yehuda Halevy - sees Christian personal and universal salvation as incompatible with the communal responsibility and national redemption
• Rabbi Menachem Kasher - sees Christian adoption of pagan practices and emphasis on afterlife as incompatible
  
• Rabbi Yosef Dov Soloveitchik - sees rejection of free will and emphasis on grace vs. human responsibility and moral choice as incompatible

Bible Search Romans-Jude for Jew with [match all forms] to see how they should be/are treated in a Christian context (esp. in Galatians). Carefully regard the meaning of "Jew first".

leads to food for thought

5.1. The Meaning and Use of Ἰουδαῖος and Ἕλλην

Scholarly interpretation of the noun Ἰουδαῖος has been quite problematic. There has been ongoing debate as to whether the term Ἰουδαῖος should be translated as ‘Jew’ in the sense of a religious identity or as ‘Judean’ in the sense of ethnic or territorial identity. Recent scholarly opinion is that the English translation of Ἰουδαῖος as Jew is an incorrect rendering and misrepresentation of the term as it is used in the New Testament. A few voices, however, dissent. Amy-Jill Levine, for instance, argues that the translation ‘Jew’ with ‘Judean’ in the New Testament leads to ‘a Judenrein (“Jew-Free”) text, a text purified of Jews. Complementing this erasure, scholars then proclaim that Jesus is neither Jew nor even Judean, but Galilean.’ As most scholars confirm the rendering ‘Judean’, they also assert that the concept is not easy to explain, since it was not clearly defined in antiquity and early Christianity. They argue instead that the term Ἰουδαῖος, translated as ‘Judean’, has different meanings. Ἰουδαῖος can be used in the context of proper name, cultural and ethnic-geographic identity, religious identity, indicator of status within the Jewish community, and pagan adherence to Judaism. Ἕλλην is used by Jews to refer to non-Jews. Here the term is used to construct a self-identity in a religious and ethnic sense to designate all who neither are born Jewish nor had become Jewish proselytes.
It can be said that the above discussion about the use of Ἰουδαῖος as either ‘Jew’ or ‘Judean’ has little to contribute to the question of the use of the pair Ἰουδαῖος and Ἕλλην in Galatians. One of the reasons is that much of the debate about the use of Ἰουδαῖος in the New Testament hinges on the problem revolving around the use of the expression in the gospels, especially the problem of the anti-Semitic tendencies found in Matthew and John. The second reason is that the group, or the communities, that made up the Ἰουδαῖος in the Matthean and Johannine Gospels are different from those that made up the Pauline communities. This said, we add that the type of Jews and Judaism that the writers of the Gospels of Matthew and John were opposing in their respective Gospels are different from the Jews and Judaism Paul is reacting to in his letters. Furthermore, assuming that the members of the Matthean and the Johannine community were Jews living within the ethnic boundaries of Palestine, then one could argue that Ἰουδαῖος is used in the Gospels of Matthew and John in a geographical sense (as well as religious), and that the supposed conflict in the passages is between Jews and Jewish Christians. On the contrary, Pauline audiences were Hellenistic Jews, that is, Jews living in the Diaspora. They are not necessarily from Palestine, but they share a Jewish identity with the rest of the Jews. In this second context, Ἰουδαῖος is used in a religious sense.
Proceeding from the above observation, a question arises: How does Paul use Ἰουδαῖος in Galatians? To answer this question, we first have to examine how Paul uses the term in the other letters, especially where Ἰουδαῖος occurs together with Ἕλλην. From the result of our study, we will proceed to show how Paul deploys the term Ἰουδαῖος and Ἕλλην in Galatians. We have few instances in Paul’s homolegoumena where Ἰουδαῖος is paired with Ἕλλην or with a related term like ἔθνος. The references are 1 Cor. 1:22–24; 12:13 and Rom. 3:9, 29 (cf. 11:17–32).
In 1 Cor. 1:22–24, Paul pairs Ἰουδαῖος with Ἕλλην in 1:22, Ἰουδαῖος with ἔθνος in 1:23, and Ἰουδαῖος with Ἕλλην in 1:24 to talk about the worthlessness of discord in the community of those who believe in Christ. The pairing of Ἰουδαῖος with Ἕλλην or other concepts related to the latter poses an important question, namely: What are the reasons for Paul’s uses of these pairs of opposites? There has been a long-standing debate about whether there was a Judaizing faction in Corinth. The implication is that if the answer is in the affirmative, then there is the possibility that there existed Jewish Christians in Corinth. Where there are no Jews, as Gordon Fee and a few others suggest, it means that Ἰουδαῖος with Ἕλλην is used in 1 Cor. 1:22–24, but not exclusively, for rhetorical purpose. Alternatively, it could be, as Fee observes, that in 1 Cor. 1:22–24 Ἰουδαῖος is universalized, as is Ἕλλην. What this means is that the terms represent the full stretch of humanity.
As we continue to deal with whether Ἰουδαῖος and Ἕλλην occur in 1 Cor. 1:22–24 with reference to the presupposed Corinthian audience, the tenets of Judaism or is a literary word to speak about the attitude and the perspective of the people, another aspect of concern is whether there is any difference between the use of Ἕλλην and ἔθνος in 1:22–24. James Scott and a few others have examined the use of Ἕλλην and ἔθνος in Pauline letters and noted that Paul is very inconsistent in his use of these terms. In their opinion, Paul uses the terms synonymously or associates each of them with multiple meanings. What this means is that Ἕλλην and ἔθνος can be used interchangeably in either a religious, ethnic, or rhetorical senses.
Given the fact that we have not studied 1 Cor. 1:22–24 in depth, we cannot claim with certainty the sense Ἰουδαῖος/Ἕλλην or Ἰουδαῖος/ἔθνος make in the context under debate. One thing is sure, in 1 Cor. 1:22–24 Paul is speaking to the Corinthian audience. Regardless of whether the terms are used rhetorically or carry religious overtones, the message of 1 Cor. 1:22–24 is addressing those who were causing division in the community, and this original audience would have found the statement clear. In that respect, the use of Ἰουδαῖος/Ἕλλην or Ἰουδαῖος/ἔθνος establishes the identity of the addressees, for in the use of the terms in question Paul is summoning the community to live in peace and avoid factions.
In 1 Cor. 12:13, the plural form of the pair Ἰουδαῖος/Ἕλλην occurs with an εἴτε … εἴτε construction, expressing the conditional sense of the message of 1 Cor. 12:13. Like in 1 Cor. 1:22–24, Paul is thinking along similar lines in 1 Cor. 12:13. The audience in 1 Cor. 1:22–24 are the same as those in 1 Cor. 12:13. If they were Jewish Christians in Corinth, the same is true for those believers to whom the message of 1 Cor. 12:13 was addressed.
What we have said of the use of the pair Ἰουδαῖος/Ἕλλην in 1 Cor. 1:22–24 resonates with the use of the term in Rom. 3:9, 29 (cf. 11:17–32). Although there are changes of context and theme, the use of the pair does not fall short of speaking about the audience. In the previous contexts, as in Rom. 3:9, 29, we do not negate that the words may have religious connotation. While there has been ongoing debate about whether Romans was addressed only to Jewish believers, Gentile believers, or both Jewish and Gentile believers, it is important to note that our understanding of who Paul’s addressees in the letter to the Romans were can help us sustain the view that Ἰουδαῖος and Ἕλλην as used in context are not concerned with geographic claims. As with the case of 1 Corinthians and Romans, Paul’s use of Ἰουδαῖος and Ἕλλην carries religious as well as ethnic connotations.
In Galatians, we also grapple with the question of whether or not the believing community had Jewish counterparts. So far, we have attested that the theological argument in Galatians was partly Paul’s response to the issues brought forward by his opponents. Behind the so-called opponents’ statement is the question of the identity of the opponents, and behind the question of the identity of the opponents is the question of the historical existence of these Jewish Christians. From Gal. 1:6–9; 4:10; and 6:12, we infer with some certainty that the Jewish Christian missionaries had visited the Galatians believers. It does not matter whether their presence is one that is short-lived or they stayed in Galatians for some time. The statements in Gal. 1:6–9; 4:10; and 6:12 do call attention to the fact that, first, there were Jewish Christians in Galatians at the time Paul was writing the letter. Second, these Jewish Christians are called Ἰουδαῖος from the point of view of their advocating circumcision and calendar observations. Third, when Paul was using the statements in Gal. 3:28a, the presence of both the Jewish Christians and the core tenet of Judaism were acknowledged.
While the designation Ἕλλην is used in the context of 3:28a to speak of the Gentile believers, it is quite doubtful that the Gentile believers in Galatia were Greeks in the strict sense of the word. The debate that Galatians was addressed to those who inhabited the original Galatian territory (North Galatia), or those who lived in the Phrygian Galatia and Lycaonian Galatia (South Galatia), could help establish the assumption that Ἕλλην in 3:28 is not used in the geographical sense. Like the use of Ἰουδαῖος, Paul’s reference to Ἕλλην could best be taken in an ethnic and religious senses. Ethnically, it refers to those who do not identify themselves as physical descendants of Abraham and Sarah (cf. the use the Abraham narrative in 3:14–29 and the allegory of Hagar and Sarah in 4:21–31). Religiously, it refers to those who do not adhere to the Jewish Law and observe certain Jewish rituals, such as circumcision (3:1–5, 27; 4:17; 5:2) and calendar observance (4:10). It is probably the idea behind what he says in Gal. 2:3 that prompted him to use Ἕλλην in 3:28a in the context of cultural and religious identities. It could also be that Paul’s Jewish background influenced him here.

The fundamental driving principle of Judaism is that ':Elohhiym's authority over the religion ended when the last prophet spoke (Malachi).

Interesting ... can you provide a source for this line of argument -- it is new to me. I've normally seen an argument along the lines of the Tanakh speaks of two offices kohen and navi. With the destruction of the 2nd Temple, the kohen sacrificial duties were obsolete (in the sense of not being able to be fulfilled). The position over time came to be referred to as rabbi. Depending upon the thread of Judaism that my friends identify with, the role of navi receives different answers.

the parable of the oven

Babylonian Talmud, tractate Bava Metzia 59a-b which is also quoted for questions regarding public vs. private revelation - an issue that often turns up in Christianity. 

I am not regurgitating a specific, particular self-made rabbinic declaration made in some tractate, although the conclusion I describe (and it is my assessment) does extract from a variety of passages found across the Talmuudh

Okay, I'll make a mental note that this is your own thoughts not an academic dispute I had not run into. Thanks for clarifying as your position is interesting. Fortunately, it is a debate that for me is of no material import -- only personal curiosity, in this case tangentially of interest re: implied position on Oral Torah. Your links were helpful.