Warm the olive oil in a large pot or Dutch oven over medium heat, then cook the kielbasa for 4 to 5 minutes until the edges brown and crisp slightly. The sausage should sizzle steadily and smell smoky, and a little rendered fat should be left in the pot.
Lift the kielbasa out with a slotted spoon and add the onion, carrot, and celery to the pot, then cook for 5 minutes. The vegetables should look softened and glossy, not browned hard, as they pick up flavor from the sausage drippings.
Stir in the garlic and red chilli flakes and cook for 1 minute more. You should smell the garlic as soon as it hits the heat, and the mixture should stay fragrant without scorching.
Add the potatoes and chicken stock, bring the soup up to a boil, then cover and lower the heat to a simmer for 10 to 12 minutes. The potatoes are ready when a fork slides in easily but the cubes still hold their shape.
Mix a ladle of the hot soup with the milk in a measuring jug or bowl, then pour that tempered mixture back into the pot and return the kielbasa to the soup. This step warms the milk gently so it blends in more smoothly instead of hitting the hot soup cold.
Add the cheddar and stir over low heat until it has fully melted into the broth. The soup should look creamy and unified, and this is the point where boiling can make the dairy split.
Simmer the soup gently for 2 to 3 minutes, then stir in the parsley and season with salt and pepper to taste before serving. It is ready when the broth lightly coats the back of a spoon and the parsley brightens the whole pot.