The Midrash[2] famously states that the supernal angels attempted to litigate against the giving of the Torah to the Jewish people for various reasons. The final rebuttal, or comeback, which closed all litigation and allowed the Torah to be given involved the Mitzvah of not eating meat and milk together. The argument was as follows:[3] After the angels told Hashem that they desired to keep the Torah for themselves, Hashem answered the angels that it states in the Torah “Thou shall not eat a kid in its mother’s milk.” Now, you angels surely remember the meal you ate in the home of Avraham Avinu? You ate meat and milk together during that meal as the verse[4] states “Vayikach Chema Vechalav,” so how can you now ask to receive the Torah? This, states the Midrash, was the final comeback which refuted any claims from the angels, and allowed the Torah to be given. Accordingly, we eat dairy and then meat on Shavuos to emphasize the reason why we received the Torah over the angels, as they did not keep the dietary laws of separating between meat and milk.[5]

A fascinating explanation of the Alter Rebbe:[6]

An even deeper approach is stated in the name of the Alter Rebbe: The law is that while one may not eat dairy after meat, one may eat meat after dairy.[7] The above Midrash is hence puzzling, as the verse explicitly states that it offered first dairy and then meat to the guests, and they therefore performed no transgression. The explanation is as follows: The reason for the prohibition against eating meat and milk together is because milk is from Chesed and meat is from Gevurah, and their combination can be catastrophic. However, this only applies if the Gevurah overpowers the Chesed, while if the Chesed overpowers the Gevurah then it is actually a positive matter. Now, we have a general rule of Tatah Gavar, the bottom overrules, and hence if one first eats dairy, he may eat meat afterwards, as the bottom which is dairy/Chesed, overrules the meat/Gevurah. However, if one eats meat first, then the Gevurah overpowers the Chesed. This system however only applies in this world, in which we hold that the lower realms are of greater importance than the higher realms, and hence the lower item overpowers. However, in the Heavens, they believe that the higher realm is greater than the lower realms, and therefore the rule of Tatah Gavar does not apply. Accordingly, Hashem told the angels that if the Torah is given to them, and thus the higher realms prevail, it would end up that they ate meat and milk together, hence transgressing the Torah. Furthermore, by the mere fact that by Avraham they agreed to eat first dairy and then meat shows that they too agree that the lower realms overpower, and hence they have no claim to receive the Torah. [Accordingly, we can explain the custom of eating a dairy meal on Shavuos, and then eating a meat meal, as this commemorates the victorious rebuttal which gave us the Torah to begin with. It also emphasizes that the purpose of the Torah is for the lower realms.]

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[1] See Beir Heiytiv 494:8 [towards end]; Toras Menachem 5743 3:1579 [brought in Shulchan Menachem 3:41]

[2] Midrash Raba Yisro 28

[3] Midrash Tehilim 8; Daas Zekeinim on Vayeira ibid

[4] Vayeira 18:8

[5] Beir Heiytiv 494:8 [towards end] “I heard that we eat dairy and then meat, unlike what the angels did by Avraham in which they ate meat and milk, as due to this the Torah was given to the Jewish people”; Toras Menachem 5743 3:1579 [brought in Shulchan Menachem 3:41]

[6] Brought in Sefer Pardes Haaretz [Horowitz] Vol. 3 p. 548 in footnote

[7] See Michaber Y.D. Chapter 89