Sear the kielbasa: Set a heavy-bottomed pot or Dutch oven over medium-high heat and add the sausage slices in a single layer, working in batches if needed. Cook for 2-3 minutes per side until the edges are dark and crispy. Remove with a slotted spoon and set aside.
Cook the vegetables: Add the onion, carrots, and celery to the same pot with the sausage drippings. Sauté over medium heat for 5-6 minutes, scraping up the browned bits, until the vegetables are soft and just starting to color.
Bloom the garlic and spices: Stir in the garlic, smoked paprika, thyme, black pepper, and red pepper flakes. Cook for about 30 seconds, stirring constantly, until fragrant.
Simmer the soup: Pour in the chicken broth, scrape the bottom of the pot to lift up any browned bits, then add the potatoes. Bring to a boil, reduce the heat to medium-low, and simmer uncovered for 15-18 minutes until the potatoes are fork-tender.
Thicken it naturally: Turn off the heat and use a potato masher to gently press the potatoes 3-4 times until about a third of them are broken down and the broth turns thick and creamy.
Finish with cream: Return the sausage to the pot, stir in the half-and-half, and warm over low heat for 4-5 minutes. Do not let it boil. Taste and add salt only if needed.
Serve: Ladle into deep bowls and top with fresh parsley and shredded cheddar if desired. Serve with crusty bread for dunking.