Preheat the oven to 375 degrees F. Prepare meatballs by combining loose ground chicken meat (discard casings, if using sausages), bread crumbs, milk, egg, parsley, and the 1/4 cup Parmigiano-Reggiano in a large mixing bowl. Season with salt and pepper, then mix until fully combined. Using a small ice cream scoop, make balls and set on a roasting tray. Drizzle lightly with oil and sprinkle with 2 tablespoons Parmigiano-Reggiano. Roast in the preheated oven for 15 to 20 minutes, until golden brown and caramelized.
While the meatballs are roasting, prepare the soup. Set a large stockpot over medium heat. Add a 2-count of oil (about 2 tablespoons), the garlic, and thyme. Gently saute until fragrant, about 2 minutes. Add carrot, onion, and celery. Season with salt and cook for 5 to 7 minutes. Pour in chicken broth and add peppercorns, the 2 tablespoons parsley, and bay leaf. Bring to a boil and then reduce heat and simmer for 25 minutes.
Once meatballs are cooked, scrape them into the pot of chicken soup and add tortellini. Bring to a boil and cook for 2 to 3 minutes to allow the flavors to come together. Remove peppercorns and bay leaf. Season with salt and peppers and serve in shallow bowls. Sprinkle with parsley and grated Parmigiano-Reggiano. Garnish each bowl with a parsley sprig and serve with some torn pieces of crusty bread.