Heat the oil in a large pot or Dutch oven over medium heat. Add the kielbasa and cook for 4-5 minutes until slightly crispy. Remove with a slotted spoon and set aside.
In the same pot, add the onion, carrot, and celery. Cook for about 5 minutes until softened.
Stir in the garlic and red chili flakes, cooking for another minute until fragrant.
Add the potatoes and chicken stock. Bring to a boil, then reduce heat and simmer for 10-12 minutes, covered, until the potatoes are tender.
Take a ladle of soup and mix it with the milk in a measuring jug or bowl to temper it. Pour this mixture back into the pot, then add the cooked kielbasa.
Add the cheddar cheese and stir until it’s completely melted.
Simmer for another 2-3 minutes until the soup thickens slightly. Stir in the parsley and season with salt and pepper to taste.
Serve the soup in bowls with extra grated cheddar on top if desired.
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