I bring my nephew to Disney every year and he has shellfish and egg allergies. In general, I have found the EPCOT restaurants able to accommodate him. My group doesn't eat at China because there is so much shrimp on the menu. We have taken him to Coral Reef, Tokyo Dining, Akershus, Rose and Crown, Marrakesh, Biergarten, all with no issues.
The only place we had a problem was Chefs de France. There is egg in practically everything there--the breads, pasta, all the desserts. Poor kid ended up having to eat a plain hamburger patty, some salad greens, and a scoop of sorbet for dessert. The restaurant folks felt bad, they just did not have the ingredients on hand to accommodate his allergy. Our waiter did tell us if we wanted we could go outside to get other food and bring it back in for him.
It s definitely worth contacting the Disney allergy specialist and letting her know which restaurants you are thinking about eating at. I probably still have her email somewhere if you need it.