The chef will come out and talk to you about the allergies, find out the "likes" of the child and then will custom make their meal for them. Disney goes out of their way to ensure that all guests are able to eat a good meal (and one that is safe or within their dietary needs).
Did your child actually get a custom meal at buffet restaurants? Neither of the 2 chefs we talked to last week asked about DD3's likes. The Ariel's chef just asked which entree she wanted and said he'd separately prepare that. But at Goofy's, the chef told us to just get the food off the regular buffet line. The reservation agent promised us it would be custom cooked, but like I said, the Goofy's chef just told me to avoid the PB&J pizza and the peanuts at the dessert bar.