**Sauté Kielbasa:** Heat olive oil or butter in a large pot over medium heat. Add the kielbasa and cook until browned on both sides, about 3-4 minutes. Remove and set aside.
**Cook Aromatics:** In the same pot, sauté the onion, carrots, and celery until softened, about 5 minutes. Add the garlic and cook for an additional minute.
**Add Potatoes and Broth:** Stir in the potatoes, chicken broth, smoked paprika, thyme, and black pepper. Bring to a boil, then reduce the heat to low. Cover and simmer for 15-20 minutes, or until the potatoes are tender.
**Prepare Cream Mixture:** In a small bowl, whisk the flour into the milk or cream until smooth.
**Thicken the Soup:** Gradually stir the cream mixture into the soup. Simmer for 5-7 minutes, stirring frequently, until the soup thickens slightly.
**Return Kielbasa to Pot:** Add the cooked kielbasa back to the soup and heat through for 2-3 minutes. Adjust seasoning with salt if needed.
**Serve:** Ladle the soup into bowls and enjoy warm.