Cook the beef in a large saucepan over medium heat, breaking into small pieces, for about 5 minutes or until no pink meat remains. Add the garlic and sauté for 1 minute. Add the pasta sauce, tomatoes, vinegar, and 2 tablespoons of the basil. Bring the mixture to a boil; reduce heat and simmer gently for 10 minutes.
Preheat oven to 350° F. In a small bowl, combine the ricotta, provolone, 2 tablespoons of the basil, the egg, and the crushed red pepper. Place 3/4 cup of the pasta-sauce mixture in the bottom of a 13-by-9-inch baking pan. Top with 3 cooked noodles, then spread a third of the ricotta mixture evenly over the noodles. Sprinkle with a third of the mozzarella and top with 3/4 cup of the pasta-sauce mixture. Repeat the layers, ending with 3/4 cup of the pasta-sauce mixture and the final 3 noodles. Spread the remaining sauce on top, sprinkle with the Parmesan, and cover with foil.
Bake for 30 minutes; uncover and bake for 15 minutes or until bubbly. Remove from oven. Let stand 15 minutes before serving. Garnish with the remaining basil.