In the same pot, add diced onion, carrot, and celery. Sauté the vegetables for 5 minutes until softened. Then, add the minced garlic, stirring for another minute until fragrant.
Add the cubed potatoes and chicken broth to the pot. Bring the mixture to a boil, then lower the heat and simmer for 15 minutes.
Place the browned kiełbasa back into the pot. Gently stir in the half-and-half and shredded cheddar cheese over low heat until the cheese is melted and the soup thickens a bit
Lastly, add the chopped parsley and season with red chili flakes and salt and pepper according to your taste. Serve hot with additional parsley on top.