We have an egg allergy too (and nut as well) and Disney is great with accommodating us, although we have found the desserts to be limited, but still exciting for our little ones (Divvies cookies or sometimes soft serve ice cream).
So we were super excited to see on another website that Babycakes' dessert is supposed to soon be available at WDW restaurants, similar to ordering cakes with your reservation.
I've been in touch with Emily at DTD Babycakes (found the contact email address through the Babycakes website) and she has my name and email address to let me know when the ordering process is all hashed out.
Here is the bulk of her email to me..
"Thank you so much for getting in touch! While we are still hashing out some details with the Disney ordering process, everything should be up and running soon! You will be able to request a cake from BabyCakes NYC along with your dining reservation! We are also offering cupcakes, doughnuts, cookies, and brownie bites to guests dining within Walt Disney World.
Are you planning a visit sometime soon? I would be more than happy to keep your name and e-mail address on file and contact you as soon as the ordering process is set up, please just let me know if you're interested!
Thanks again for getting in touch! We are very excited to be offering our desserts at Walt Disney World restaurants."
So I still don't have all the details or exactly how it works or how much it costs, but I am hoping it will be available for our trip in November. Having a family dessert at Disney would be pretty exciting for us.
Hope this helps.