Reb Noson z"l expounds on this halacha through the lens of Torah 19 in LM in an intriguing way. He writes in Likutei Halachos, Yoreh De'ah: Hilchos Beitzim 2:1:
It is written in the Tikkunei Zohar (Hakdamas Tikkunei Zohar, daf 1) about the passuk "And the mother rests upon the chicks or on the eggs" (Devarim 22:3). "Eggs" are associated with the written Torah. It also says that that eggs represent the order of angels known as 'Ofanim', and that they also represent the world of Asiya. There is no useful sign to discern the eggs of a kosher bird from those of a non-kosher bird. It is known that kosher signs and non-kosher signs are associated with the holiness of Yisrael and the impurities of the Akum respectively. The pure, or kosher signs are of the side of Yisrael, which is the side of holiness. And, impure signs are from the side of Akum, which is the side of defilement. Therefore, the nation of Yisrael, which is holy and is separated from the nations, must also be separated from all impurity, which is the side of defilement.
Reb Noson continues,
Eggs which are associated with the written Torah, are also associated with non-Jews because they use the written Torah in their own capacity. Therefore there's no real sign to show that use of the written Torah implies purity (since both Jews and non-Jews have the written Torah). In regards to the written Torah it would be difficult to discern Jews from non-Jews. The only way to distinguish Jews from non-Jews with respect to the Torah is through the oral Torah. Through the oral Torah the Jew is separated from the non-Jew and is recognizable. However, because the non-Jews have a minimal grasp on the written Torah, it cannot be used as a sign to distinguish Jews from non-Jews, as mentioned above. As said before, eggs are a concept of the Torah. The grasp that non-Jews have on the Torah is a concept of the 24 species of non-kosher birds. Birds are associated with souls. And, the 24 species of non-kosher birds are associated with the souls of Akum. The souls of Akum and the 24 impure species of birds are then associated with the 24 books of the written Torah, which are holy.
It's very interesting that Reb Noson makes the connection with the wording of the Shulchan Aruch to describe features of an egg and the nature of the connection that non-Jews have to the Torah. The word "round" as used by the Shulchan Aruch is "כד". The gematria of kad is 24, ie 24 books in the written Torah. Thus, if an egg only has round sides it cannot be kosher, because it is completely written Torah.
Reb Noson continues,
The birds that are pure and kosher are not enumerated in the written Torah whatsoever (only the impure bird species are written in the Torah), because pure birds, which are a concept of the souls of Yisrael, have an advantage. This advantage is that they also have the oral Torah in addition to the written Torah. Therefore, the kosher species were not written in the Torah. Rather, they were passed on through tradition. Our Sages said: One can only eat a bird which is known to be pure through tradition (Chulin 63). This is because the essence of eating kosher bird species in through tradition, ie through oral Torah.
Reb Noson's connections are genious! If an egg only has rounded (kadin) or sharp (chadin) sides then it can't be kosher! If only round, then it only possesses written Torah. If that's the case then it's impossible to know if it's from a kosher animal anyway, because the written Torah doesn't tell us which birds are kosher. And, on the same note, if both sides are pointed, meaning if there is ONLY oral Torah and interpretation without written Torah, then that too is invalid. For an egg to be kosher it needs both a round and pointed side. For a Jew to keep Hashem's Torah correctly he needs both the written and Oral Torah, which were both given to Moshe Rabbeinu on Har Sinai. Furthermore, even if an egg have both features, one still must find out from another person if it actually comes from a kosher specie. This is the "tradition" aspect. There are times when a person still needs mesorah to determine something's validity in spite of both rounded and pointed edges.
If you take a look at this halacha in LH, you'll read further that Reb Noson continues to explain the importance of why an egg (or a Jew) needs both a pointed and rounded side. The appearance of both a lion and the Bais HaMikdash are larger on one side and narrow on the other.
I can continue this Torah if anyone wants.