HEAT A LARGE Dutch oven over medium heat and add the olive oil. When the oil is hot, add the kielbasa and cook and stir until browned all over, about 5 minutes. Add the leeks and thyme and cook until the leeks are wilted, about 6 minutes. Add the garlic, paprika, allspice, and red pepper flakes and cook until fragrant, about 1 minute. Add the tomatoes, chicken broth, and bay leaves and season with salt and pepper. Bring to a simmer, cover, and cook 30 minutes to develop the flavors.
Add the potatoes and kale. Simmer, uncovered, to reduce the soup and concentrate the flavors a bit, until the kale and potatoes are very tender, about 30 minutes. Remove the bay leaves and serve the soup with a final drizzle of olive oil and some grated Parmesan.