Heat the oven to 375°F and grease a 9-by-13-inch baking dish with 1 teaspoon of the oil.
Brown the beef, pork and onion in a deep skillet over medium-high heat for 8 to 10 minutes, until the meat reaches 160°F, then drain the fat.
Add the remaining oil and cabbage and cook for 8 minutes, stirring often, until the cabbage begins to wilt.
Stir in the crushed tomatoes, tomato sauce, broth, rice, brown sugar, salt and pepper.
Transfer the mixture to the dish, cover tightly with foil and bake for 55 minutes.
Stir from the edges toward the center, cover again and bake for 10 to 15 minutes more, until the rice and cabbage are tender.
Rest the covered casserole for 10 minutes before serving.