Step 1: Place the turkey carcass into a large stock pot. Add chopped carrots, celery, quartered onion, thyme sprigs, rosemary sprigs, bay leaves, and black peppercorns. Cover with cold water.
Step 2: Bring to a gentle boil, then reduce heat to a low simmer. Skim off any foam that rises. Simmer uncovered for 2-4 hours, or longer, for a rich stock.
Step 3: Carefully remove the turkey carcass and large vegetables. Strain the broth through a fine-mesh sieve or cheesecloth into a clean bowl. Discard solids. Cool and refrigerate the stock, then skim off any solidified fat.
Step 4: In a large Dutch oven or soup pot, heat olive oil or butter over medium heat. Add finely diced carrots, celery, and onion. Sauté for 5-7 minutes until softened and translucent.
Step 5: Pour the homemade turkey stock into the pot. Stir in the shredded or diced cooked turkey meat. Add salt and fresh black pepper to taste.
Step 6: Bring the soup to a gentle simmer. Add the small pasta or rice and cook according to package directions, typically 8-12 minutes, until al dente.
Step 7: Taste the soup and adjust seasonings as needed. If using, stir in fresh spinach or kale during the last few minutes of cooking until wilted.
Step 8: Ladle the hot soup into bowls and garnish with fresh parsley or a sprinkle of Parmesan cheese, if desired. Serve immediately.