Pizza Port in Tomorrowland, a counter service place, is a big favorite of my boys, ages 10 and 11. None of our family is a fan of the pizza, but the pasta dishes are good, particularly the tomato basil and chicken fusilli.
We did try the Boardwalk Pizza and Pasta place in the Paradise Pier area of the Disney California Adventure park last month and really enjoyed it. This place can compete with Pizza Port for my boys' taste buds. The Boardwalk place has tasty flatbread pizza and good pasta dishes. This also is counter service.
If you like fried chicken, you can get that at the Plaza Inn, and the portions are big and easily shareable. My boys like the chicken and potatoes but not the green beans. You can ask for a double portion of potatoes though and skip the beans, so buying one plate done up this way works for a shared meal for my sons. Then they still have room for dessert. If you're looking for a different kind of grease, there's the Little Red Wagon corn dog cart right outside the Plaza Inn. You can get your corn dog with chips, apple slices, or a la carte.