Preheat the oven to 350 degrees F (175 degrees C). Oil the bottom and sides of two rectangular glass baking dishes.
Melt butter in a 4- to 6-quart stockpot over medium heat. Add ground beef, green and red cabbage, onion, celery, and garlic. Stir in diced tomatoes, Worcestershire sauce, salt, oregano, basil, and pepper. Cook until beef starts to brown and cabbage softens a bit, 8 to 10 minutes. Stir, and continue cooking for 10 to 15 minutes.
While the beef cooks, prepare bell peppers: remove stems and cut peppers in half from top to bottom. Remove seeds and white membrane. Rub outsides of the peppers with olive oil and arrange, skin-sides down like edible bowls, in the prepared baking dishes.
Strain beef mixture to remove excess liquid; scoop into peppers to fill and mound any extra on top or reserve for another use. Cover with foil.
Cook in the preheated oven until peppers have softened, 30 to 40 minutes. Remove foil, top with mozzarella and provolone, and cook for an additional 5 minutes.
Remove from the oven and allow to cool for a few minutes before serving.