And only have a minute (due to an extremely irritable, itchy 4 month old), but I want to say that your allergist is wrong wrong wrong that eggs cooked "in" something are okay. My now-3 y/o used to react to my breastmilk when I ate things with egg cooked "in" them. An egg allergy can be so severe that even trace amounts cause problems. I'd eliminate all traces of egg immediately.
johng gave you some great advice and take a good look at aunty's replies below. Good luck.