"Don't believe me? Ask the dishes!" -Beauty and the Beast
Picture this: you're at a restaurant fit for royalty, the food is expensive, a princess is due at your table any minute... but you have no idea what to order.
If all of your options are looking like bad options at a table service restaurant where you can't order a Mickey Check kids meal, don't freak out. Disney is extremely accommodating and if you have any questions about what the chef can or can't do; just ask your server and they'll get someone to ask the chef for you.
For example, my family and I had a reservation at Be Our Guest and when we went to order, we were set up with a server who could contact the chef and see what they could arrange. On the kids menu there is a grilled shrimp and broccoli option, so we were able to order that but in a larger, adult size (plus some mashed potatoes without butter to go with it) after asking if it was possible. Our meal turned out to be delicious, easy to track, and we weren't worried the whole day about how much we ate!
This can work at any restaurant - don't be shy. Disney prides itself on good customer service and they understand that everyone has different dietary needs and restrictions.