Heat the olive oil in a large pot or Dutch oven over medium heat. Add the sliced kielbasa and cook until lightly browned, about 5 minutes. Remove the sausage with a slotted spoon and set aside, leaving the rendered fat in the pot.
Add the chopped onion and celery to the pot. Cook until softened, about 5 minutes.
Stir in the minced garlic, smoked paprika, thyme, and black pepper. Cook for 1 minute until fragrant.
In a small bowl, whisk the flour with 1/2 cup of the chicken broth until smooth. Pour this slurry into the pot, stirring constantly.
Add the remaining 3.5 cups of chicken broth and the diced potatoes. Bring the mixture to a boil, then reduce heat, cover, and simmer for 10 to 12 minutes, or until the potatoes are tender when pierced with a fork.
Stir in the milk and heavy cream. Heat through gently, but do not allow the soup to boil after adding the dairy.
Return the cooked kielbasa to the pot. Taste and add salt as needed.
Serve hot. This is a great family favorite soup recipe.
Notes
For a thicker soup, mash about 1 cup of the potatoes against the side of the pot before adding the cream.
You can substitute smoked sausage for kielbasa if needed.
This recipe works well for meal prep; store leftovers in an airtight container in the refrigerator for up to 4 days.