I always had a hard time eating eggs(plain) so I'm not shocked. I don't think I really thought about all of the underlying things that could have eggs in it.
Maybe not with your egg sensitivity, but I've noticed I'm sensitive to milk/cream but only if it hasn't been cooked. It's not lactose intolerance, it's a whey sensitivity. So I can eat almost any cheese, since the main protein there tends to be casein. Greek yogurt is fine as well, somehow. I'm totally fine with baked goods or dishes when they call for some milk or cream, but milk shakes, ice cream and whey protein bars will get me; it's like a mild anaphylaxis -- tongue gets itchy and I know to stop. Luckily, it's not serious.