Brown sausage and onion in a 6-quart Dutch oven over medium-high heat for 8 minutes, breaking the sausage into small crumbles and cooking it to 160°F. Drain excess fat.
Stir in garlic and tomato paste for 1 minute. Add crushed tomatoes, broth, oregano, salt and pepper and simmer for 8 minutes.
Add broken noodles and simmer uncovered for 12 to 15 minutes until tender, stirring frequently during the first 5 minutes to separate the pieces.
Mix ricotta and Parmesan in a small bowl while the noodles cook.
Divide soup among 6 bowls and top each with one-sixth of the ricotta mixture and mozzarella. Let the cheese soften in the hot broth before eating.