We were there this past december. The 3rd time for me since diagnosed with an egg allergy. I am also allergic to peanuts as well.
At the counter service, tell the cashier when you go to place your order that you have a food allergy in the group. They always sent out a manager with an ingredient binder and I was able to find out what I could eat. I was allowed to make substitutions for things that were not safe.
Ex: at flame tree BBQ, I cannot eat corn bread or coleslaw so they gave me an order of fries instead and they did not charge me.
For desserts, the issue wasn't always eggs, it was the cross contamination of peanuts. I was always allowed to substitute either fruit or if they had allergy safe desserts, I could get one of those.
We were there for 6 days and we had the QS dining plan and I was never dissapointed. I always had more than enough options.
And if you are not on the dining plan, you are not limited to just the kids options at CS.
My advice, bring snacks with you to the park. If you are going to WDW during a busy time, make sure you go to the CS places either right when they open or afte rthe rush (after 1PM or so) because it will be less crowded and faster. But that's hwy I say bring snacks with you so that if she is hungry and it is crazy busy, you have something to hold her over until it empties out a little bit or while waiting for the food.
We ate at (the on'es I can remember at least):
EPCOT:
Sunshine Seasons
Tangierin Cafe (Morroco)
MK:
Cosmic rays
Caseys Corner
Village haus
Pecos Bills
DHS:
Studio Catering Company
AK:
Flame Tree BBQ
All had plenty of options that were egg free for me. And we had excellent service at all of them