Prepare the Kielbasa and Vegetables: In a spacious saucepan, warm the butter over medium heat. Once melted, introduce the sliced kielbasa and sauté until it's nicely browned, which should take approximately 5 to 7 minutes. Afterward, take the kielbasa out of the pot and set it aside, keeping the rendered fat in the saucepan.

Cook the Vegetables: Using the same pot, toss in the diced onion, chopped carrots, and celery. Cook these vegetables for about 5 to 6 minutes, stirring occasionally, until they become tender.

Add Potatoes and Broth: Add the minced garlic and continue cooking for an additional minute, allowing the aroma to develop. Next, incorporate the cubed potatoes and add the chicken broth. Increase the heat until the mixture reaches a boil, then lower the heat to let it simmer for 15 to 20 minutes, or until the potatoes are soft and easily pierced with a fork.

Create the Creamy Base: In a separate bowl, combine the flour, milk, and heavy cream, then use a whisk to blend the ingredients until the mixture is free of lumps and creamy. Gradually stir this mixture into the pot, making sure to avoid lumps. Simmer for another 5 minutes, allowing the soup to thicken slightly.

Season and Add Kielbasa: Return the cooked kielbasa to the pot and season the soup with salt, pepper, smoked paprika, and thyme (if using). Stir well and cook for another 5-7 minutes to let the flavors blend.

Finish and Serve: Once the soup is thickened and flavorful, ladle it into bowls. Garnish with shredded cheddar cheese and fresh parsley if desired. Serve hot and enjoy!