In a large pot or Dutch oven, brown the ground beef or turkey over medium heat until fully cooked, breaking it into smaller pieces with a spatula. Drain excess fat if necessary.
Add the diced onion, minced garlic, carrots, and celery to the pot. Sauté for about 5 minutes until vegetables are tender.
Stir in the crushed tomatoes, chicken or vegetable broth, dried basil, dried oregano, salt, and pepper. Bring the mixture to a boil.
Once boiling, reduce the heat to a simmer. Add the gluten-free pasta and cook according to the package instructions, usually about 8-10 minutes, until al dente.
While the pasta cooks, in a small bowl, stir together the ricotta cheese, half of the mozzarella cheese, and half of the Parmesan cheese until well combined.
When the pasta is done, remove the pot from heat. Stir in the ricotta mixture until incorporated.
Serve the soup hot, topped with the remaining mozzarella and Parmesan cheeses, and garnish with fresh basil leaves if desired.