Heat the oven to 350°F and simmer the whole cabbage in a large pot of water for 8 minutes, peeling off 12 softened outer leaves as they loosen.
Trim the thick center rib from each leaf without cutting all the way through.
Mix the beef, rice, onion, egg, salt and pepper gently and divide the filling among the leaves.
Fold the sides over the filling, roll each leaf snugly, and arrange the rolls seam-side down in a 9-by-13-inch baking dish.
Whisk the tomato sauce, broth, brown sugar and vinegar, pour it over the rolls, and cover the dish tightly with foil.
Bake for 90 minutes, uncover for 10 minutes, and check that the filling reaches 160°F before serving.