Steam or microwave the cabbage leaves until tender. Let cool. Cut out a 2- to 3-inch wedge of the tough stalk from the bottom of the leaves, taking care to leave the rest intact. Set aside.
Coat the bottom of a large Dutch oven with olive oil and place over medium heat. Add onions and toss to coat. Cover and sweat the onions, stirring occasionally, until onions are soft and translucent. Add tomatoes, quartered garlic cloves, and water. Simmer about 15 minutes, stirring occasionally.
Meanwhile, combine ground beef, Italian sausage, diced onions, salt, pepper, oregano, basil, pressed garlic, egg, tomato paste, and rice. Divide meat mixture into 8 portions.
Place each meat portion at the curled tip of a cabbage leaf. Fold the top of the leaf down over the stuffing, fold the sides of the cabbage leaves in toward the center, and roll up to enclose the filling.
Place cabbage rolls seam-side down in a deep, heavy skillet or Dutch oven. Top with the tomato sauce. Cover and simmer over medium-low heat about 1 hour.