Start by melting butter in a large pot over medium heat. Add the chopped onion, diced carrots, and celery. Stir occasionally and let them cook for around 5 minutes until they become soft and fragrant. This base layer builds the savory flavor foundation for the soup.
Add the peeled and diced potatoes into the pot. Pour in the chicken broth, ensuring all the ingredients are submerged. Bring it all to a boil, then reduce the heat to a gentle simmer. Let it cook for about 15 to 20 minutes, or until the potatoes are soft enough to pierce with a fork.
Once the potatoes are tender, stir in the pre-cooked chicken. Slowly pour in the heavy cream (or milk if using), and mix well. Season the soup with salt and freshly ground black pepper. Let it heat through for an additional 5 minutes to ensure everything is warm and flavorful.
Ladle the soup into bowls while it’s hot. Add your favorite toppings like chopped green onions, shredded cheese, or fresh parsley to enhance the flavor and add visual appeal. Serve with crusty bread or on its own for a soul-satisfying meal.