Place the ground beef and sausage in a 4-quart soup pot over medium-high heat. Cook while breaking the meat into crumbles. About halfway through cooking, add the onions and continue cooking until they soften, about 5 minutes. Drain off any grease. Stir in the garlic and cook for 1 minute.
Add 4 Tbsp. tomato paste, 1 tsp dried oregano, 1 tsp dried basil, 1 tsp sugar, 1 tsp hot sauce, ½ tsp salt, ¼ tsp pepper, and a pinch of red pepper flakes. Mix everything together thoroughly.
Pour in the chicken broth. As you stir, use a silicone spatula to scrape the bottom and sides of the pot and loosen anything that has stuck to the surface.
Stir in the crushed tomatoes, undrained diced tomatoes, and bay leaf. Bring the soup to a boil, then lower the heat so it simmers.
If you have a Parmesan rind available, place it in the pot during the simmering stage for additional flavor.
Keep the soup at a gentle simmer while bringing a separate pot of salted water to a boil for the lasagna noodles. Before cooking the noodles, taste the soup and adjust the seasonings if necessary. Keep in mind that the flavor will become more concentrated as the soup continues to simmer.
Break the lasagna noodles in half and cook them according to the package instructions. Drain, then cut the noodles into smaller strips. Add them to the soup. Alternatively, place the noodles directly into individual serving bowls. If you expect leftovers—particularly if you intend to freeze the soup—keeping the noodles separate and cooking fresh noodles when serving is preferable.
Take out the bay leaf. Divide the soup among bowls and top each serving with Parmesan, Mozzarella, Ricotta, and parsley.