Add just enough oil to a large, heavy-bottomed soup pot to lightly coat (about 2-3 teaspoons), and place over medium heat. Add the sausage and cook until browned, crumbling as you go.
Add the onion to the sausage and sauté for 2-3 minutes or until beginning to soften. Stir in the carrots, celery, garlic, Italian seasoning, and crushed red pepper. Sauté for a minute or until the garlic is fragrant but not browned. Add the tomatoes with their juices and cook for another minute or two. Then stir in the chicken broth, lentils, and potato.
Increase the heat in order to bring the mixture to a boil, then partially cover the pot and reduce the heat to maintain a gentle simmer. Cook for approximately 40 minutes, or until the lentils are tender but still hold their shape. Stir every 10-15 minutes or so.
Stir in the chopped greens, salt and balsamic vinegar. Cook for 3-4 more minutes, or until the greens have softened to your liking.
Ladle into individual bowls and sprinkle with Parmesan cheese.