Heat the pancetta in a large sauté pan over medium-high heat, stirring until it is lightly browned and its fat is released, about 2 minutes. Add the leeks, carrots, celery root, and kale. Reduce the heat to medium, cover, and cook until the vegetables are soft, about 5 minutes.
Add the chicken broth and V8; bring to a simmer for 5 minutes. Season with salt and pepper and add the basil.
Ladle the soup into 4 soup bowls and top each bowl with the Parmigiano.