Heat the olive oil in a large Dutch oven or heavy-bottomed pot over medium heat. Add the diced onion, carrots, and celery, cooking them for about 5 to 7 minutes until the onions are translucent and the vegetables begin to soften. Stir in the minced garlic and cook for an additional minute until it becomes fragrant.
Add the ground turkey or beef to the pot, breaking it up with a wooden spoon as it cooks. Continue cooking until the meat is fully browned and no longer pink. If there is an excess of fat in the pan, you can carefully drain it off before proceeding to the next step to keep the soup light.
Pour in the diced tomatoes with their juices and the broth. Stir in the dried oregano, basil, smoked paprika, salt, and pepper. Bring the liquid to a gentle boil, then reduce the heat to low. Cover the pot and let the flavors meld together for about 15 minutes.
Turn the heat back up to medium and stir in the small pasta. Cook the soup for about 8 to 10 minutes, or until the pasta is al dente. During the last 2 minutes of cooking, stir in the fresh spinach or kale. Once the greens have wilted and the pasta is tender, remove the pot from the heat and serve immediately with a sprinkle of cheese.
