In a premium Dutch oven or suitable soup pot, heat the extra virgin olive oil over medium-high temperature. Once the oil has reached the optimal temperature, introduce the kielbasa segments. Allow them to sear for approximately 5-6 minutes, ensuring regular stirring to achieve uniform browning. Once done, carefully remove the kielbasa using a slotted spoon to drain excess oil.

To the same pot, introduce the minced garlic, finely diced onions, carrots, and sliced celery. Allow these ingredients to sauté, stirring occasionally, for a duration of 2 minutes.

Gradually introduce the chicken stock followed by the diced potatoes to the pot. Elevate the heat to achieve a boil, subsequently reducing the temperature. Secure the pot with a lid and allow the ingredients to simmer for a span of 15 minutes, or until a fork can easily penetrate the potato cubes.

Once the potatoes are adequately tender, reintroduce the seared kielbasa to the pot. Additionally, pour in the milk and introduce both the cheddar and parmesan cheeses. Season with kosher salt and freshly ground black pepper. Reduce the heat to low and consistently stir the mixture for an additional 5-10 minutes, ensuring the cheeses have completely melted into the soup.

As a finishing touch, integrate the freshly chopped parsley into the soup. Stir once more before serving, ensuring a uniform blend. The Potato Kielbasa Soup is now ready to be presented and enjoyed.