What is the source and logic for eggs falling into the category of Bishul Akum? It can be eaten raw and I don’t think they would serve an omelet at a WH dinner.
Eggs are normally cooked, and therefore although they can theoretically be eaten raw, they are included in the prohibition. Tosafos (Yevamos 46a) explain that the reason for this is that eating them raw is not an “important” manner of consumption (see Avodah Zarah 38b where there is a dispute concerning eggs; the halachic ruling is that they are included in the prohibition).
They are also served at “important meals” (even at aristocratic tables) in various ways (this will include an omelet, which is certainly served for breakfast), and therefore they are included in the prohibition.