There is the Pizza Port in Tomorrowland that sells pizza. In Downtown Disney, there is a counter shop called Napolini, where you can get pizzas, pasta, etc., and if you like a nice sit-down restaurant, you have Naples Ristorante. In the Storyteller Cafe in the Grand Californian Hotel, you can order "make your own pizza" and that might be fun for him.
As for cheeseburgers - I know they sell them at Hungry Bear Restaurant in Critter Country at Disneyland. They also have some at Whitewater Snacks in the Grand Californian Hotel. And as AFMom said, they sell them at Taste Pilots Grill, in California Adventure.