We have had really great experiences at most Disney restaurants both TS and QS for our multiple food allergies including egg. Some standouts are 50s PT in DHS. Their gluten free, egg free, peanut/tree nut free fried chicken is AWESOME! Other great TS are Raglan Road, The Wave, Frontera Cucina, Jungle Skipper Canteen, and 1900 Park Fare. Some great QS are Pecos Bills, Yorkshire Fish, Satu'li Canteen, Pizzafari, and most resort QS we have visited.
The worst experiences we have had by far were both at CSR which brings me to my next point. Be extra careful and take your patience along if you visit the restaurants that are run by outside vendors such as those at CSR and many in Epcot WS. We have found their allergy options are often very limited and there seems to be a more casual approach that never gives me a feeling of confidence. My daughter was served egg noodles at Pepper Market after I questioned the server several times because the noodles were different from our previous visits. Needless to say, after my daughter took a tiny bite, I made her spit it out after my mama bear instincts kicked in. I took the plate back to the station and demanded to see the chef manager who confirmed my fears.