For individual portions, place small, shallow baking dishes (each holding about 2 1/2 to 3 cups) on a rimmed baking sheet. Divide the pasta mixture among them; top with cheese, then bake and broil.
Preheat oven to 400 degrees. Heat oil in a straight-sided skillet over medium-high. Add meatballs and cook, turning occasionally, until browned all over, 8 minutes. Add garlic and cook, stirring, 45 seconds. Add pureed tomatoes and bring to a boil, then reduce heat and simmer 25 minutes.
Meanwhile, cook pasta in a large pot of boiling salted water until al dente, about 2 minutes less than package instructions. Drain pasta; return to pot and toss with sauce and cubed cheese.
Transfer mixture to an 8-inch square baking dish; cover with parchment-lined foil. Bake until bubbling, 15 minutes. Uncover; top with sliced cheese. Adjust oven to broil; cook until cheese is melted and golden brown, about 2 minutes. Let cool 10 minutes and serve.