I see that you have posted lots of questions about specific places and their handling of allergies.
Rest assured, Disney has very specific protocols that they follow when dealing with allergy orders, no matter where in Disney the restaurant is. If something on a menu cannot be made safe for your daughter, the chefs will make her something not on the menu.
I have 10+ foods that I cannot eat due to severe gastrointestinal symptoms (pain, nausea, vomiting). None of my reactions is anaphylactic, though. On our recent nine day trip, I did not get sick once.
I know how scary eating out can be. I can't imagine how much scarier it is when anaphylaxis is a possible reaction. However, your daughter will be very well taken care of!
At the food court, just like at a QS restaurant, you will ask to speak to the chef. The chef will take down your info and they will have it for the remainder of your visit. We lucked out that every time we ate at the food court at AoA, the same chef was working. Your daughter's order will be taken by the chef or a manager and her order will be brought out by the chef or manager.
Have a great trip!