Spread cabbage in the bottom of a 9x13-inch baking dish; sprinkle rice on top.
Heat a large skillet over medium heat. Cook and stir beef in the hot skillet until browned and crumbly, 5 to 7 minutes; drain and discard grease. Add onion, oregano, basil, garlic powder, and thyme; cook and stir until onion is transparent, about 5 minutes. Season with sugar, salt, and pepper.
Spoon beef mixture over rice and cabbage in the baking dish. Break up tomatoes in the can with your hand; pour over beef mixture. Cover baking dish with aluminum foil.
Bake in the preheated oven until rice is tender, about 1 hour 15 minutes. Uncover and arrange American cheese slices over casserole. Let stand until cheese melts and absorbs juices, 15 to 20 minutes.