Preheat your oven to 350F with a rack in the middle position.
Prepare all of your ingredients, making sure your rice is cooked and all your vegetables are chopped, minced or grated.
Place a large Dutch oven on medium-high heat. Drizzle the olive oil in the pan and add your onion and garlic. Cook for 2-3 minutes.
Add the ground meats to the Dutch oven. Bread it down and cook it, almost all the way through.
Add the shredded cabbage and the salt to the pan. Stir it all together and cook for 5 minutes, or until the cabbage has softened.
Continue by adding the diced tomatoes, tomato sauce, grated carrots, brown sugar, paprika, Worcestershire sauce, rice, and fresh dill to the Dutch oven. Stir to combine everything very well.
Transfer the contents of the Dutch oven into your prepared 9"x13" baking dish. Level out the filling, and then sprinkle the grated cheese evenly over the dish.
Place the dish, uncovered, into your preheat oven. Bake for 20-25 minutes, or until lightly golden on top. Remove from the oven and let it rest for 5-10 minutes. Sprinkle with more freshly chopped dill and serve!