Preheat oven to 375°F. Grease a 9×13-inch baking dish.
Warm olive oil in a large skillet over medium-high heat.
Brown the ground beef, breaking it up as it cooks, about 6–8 minutes.
Stir in onion and cook until softened, 3–4 minutes. Add garlic and cook 30 seconds.
Add crushed tomatoes, tomato sauce, tomato paste, broth, and Worcestershire. Mix until smooth.
Season with paprika, oregano, dill, salt, and pepper. Simmer 10 minutes.
Meanwhile, sauté cabbage in a large pot with a splash of water until wilted, 5–7 minutes. Drain well.
Combine meat sauce, cooked rice, and half the cabbage in a large bowl.
Spread mixture into the baking dish. Top with remaining cabbage and cheese.
Cover with foil and bake 35 minutes. Uncover and bake 10–15 minutes more.
Rest 10 minutes before serving. Garnish with parsley.