Strip the kale leaves from their stems and tear the leaves into large pieces. Discard the stems. Rinse the leaves in a large bowl of cool water. Transfer to a colander and set aside.

Bring a large pan of water to a boil. Pierce each sausage link several times with a paring knife, place the links in the boiling water, and cook for 10 minutes. Transfer to a plate. When the sausages are cool enough to handle, remove the casings and cut the links into 1⁄4-inch-thick slices.

Heat the olive oil in a large soup pot over medium heat. Add the onion, celery, and carrot and cook for 5 minutes, or until the onion is soft. Add the garlic and cook for 1 minute more. Add 3 cups of water plus the stock, kale, and sausage. Bring to a simmer, stirring occasionally. Add the potato, tomatoes with juice, tomato paste, salt, basil, thyme, and pepper. Simmer for 20 to 25 minutes, or until the kale is tender. Add the beans and simmer for 5 to 10 minutes more. Taste and adjust the seasonings, if desired. Serve hot and pass the Parmesan at the table.