The Plaza is the only place inside the park to have a breakfast 50 mins before rope drop. It is really fun to do first thing in the morning. We are at the turnstyles by at least 10 mins before the park opens for MM and go right to the Plaza when they let everyone in. We eat fairly quickly I guess, but it is not crowded at all first thing, and we are able to meet all the characters, most more than once, take pics with them, and make more than 1 trip to the buffet. I love it there. We are all adults also. It's fun for all!
But if you don't want to eat in the park, most of the Downtown Disney restaurants serve breakfast, or there is a Denny's or Ihop on Harbor.
I think you should try the Plaza! It's fun!