Preheat the oven to 375°F / 190°C.
Bring a large pot of water to a boil. Carefully cut around the core of the cabbage with a sharp knife.
Place the cabbage head in the boiling water. Cook for 3 to 5 minutes, then use tongs to gently remove softened outer leaves. Repeat until you have 12 to 14 large leaves.
Lay the cabbage leaves on a clean work surface. Use a small knife to shave down the thick rib at the base of each leaf so it lies flat and is easier to roll.
Bring a small saucepan of water to a boil. Add the rice and cook for 7 minutes, until partially cooked. Drain well and let cool slightly.
In a large bowl, combine the par-cooked rice, ground beef, diced onion, minced garlic, egg, dill, salt, black pepper, and paprika. Mix gently until evenly combined.
Heat 1 tablespoon olive oil in a saucepan over medium heat. Add the onion and grated carrot. Cook for 5 minutes, until softened.
Add the garlic and cook for 30 seconds. Stir in tomato sauce, tomato paste, broth, paprika, sugar, salt, and black pepper. Simmer for 5 minutes, then remove from the heat.
Spread about 1 cup of tomato sauce in the bottom of a large Dutch oven or 9-by-13-inch baking dish.
Place about 3 tablespoons of filling near the base of one cabbage leaf. Fold the bottom over the filling, fold in the sides, and roll tightly. Repeat with the remaining cabbage leaves and filling.
Arrange the cabbage rolls seam-side down in the prepared dish. Place them close together so they stay closed while baking.
Pour the remaining tomato sauce evenly over the cabbage rolls. Cover tightly with a lid or aluminum foil.
Bake for 60 to 70 minutes, until the cabbage is tender, the rice is fully cooked, and the filling is cooked through.
Let the cabbage rolls rest for 10 minutes before serving. Spoon tomato sauce over each portion and serve with sour cream and fresh dill if desired.