Place a 5-quart Dutch oven over medium-high heat and add olive oil. Sauté kielbasa slices for 6 minutes or until caramelized and browned on both sides; remove sausage with a slotted spoon and reserve on a side plate.
Reduce heat to medium. Incorporate butter into the remaining rendered fat. Once melted, add onion, carrots, and celery. Sauté for 5 minutes until the mirepoix is softened and the onions are translucent.
Add minced garlic, smoked paprika, and dried thyme. Sauté for 60 seconds until fragrant.
Sprinkle all-purpose flour over the vegetables and stir constantly for 2 minutes to cook out the raw flour flavor while avoiding scorching.
Deglaze the pot by gradually whisking in the chicken broth, ensuring all browned bits (fond) are lifted from the bottom. Add cubed potatoes.
Increase heat to bring the mixture to a boil, then immediately reduce to medium-low. Cover and simmer for 15 minutes or until potatoes reach fork-tender consistency.
Whisk in heavy cream and reintegrate the reserved browned kielbasa. Continue to simmer uncovered for 4 minutes to achieve the final emulsion and desired thickness.
Adjust seasoning with salt and pepper to taste. Garnish with fresh parsley and serve immediately at an internal temperature of at least 165°F (74°C).