The Torah says that one should not cook a calf in its mothers' milk. This is the basis for the prohibition of eating milk and meat together. If this is the case, why is it not also illegal to cook a chicken with eggs, as it is frequently used for breading (e.g. in schnitzel)?
The Torah's prohibition:
To avoid confusion, the rabbis of the Talmud made the general rule:
So the simple answer is -- "because the rabbis didn't ban it." Presumably they were concerned that chicken-in-milk would get confused with beef-in-milk, but didn't feel that eggs looked anything like milk and thus chicken-in-egg wouldn't cause any problems.
Ask explained on Chabad.org, there are multiple rationales provided for the mitzwah of separating meat and milk, which is ultimately regarded as a hoq (Divine decree):
Personally, these latter two sources may help address why chicken and eggs may be mixed while meat and milk may not. While the egg is effectively a life form in development, it is 1) less representative of Hesed than milk and 2) chicken and eggs do not resemble closely enough the ancient 'Avodah Zarah cited by HaRaMBa"M.