Copycat Dolly Parton's Stampede Soup

Sauté the vegetables: In a large pot or Dutch oven, melt the butter over medium heat. Add the finely diced carrots, celery, onion, and potatoes. Sauté for 5-7 minutes, until the vegetables begin to soften.
Make the roux: Sprinkle the all-purpose flour over the sautéed vegetables. Stir continuously for 1-2 minutes to cook the flour and create a roux. This will act as the thickener for your soup.
Add the liquids: Slowly whisk in the chicken broth, a little at a time, to prevent lumps. Bring the mixture to a gentle boil, then reduce the heat to a simmer. Let it cook for 10-15 minutes, or until the potatoes are tender.
Finish the soup: Stir in the heavy cream and whole milk. Heat the soup through, but do not let it come to a full boil after adding the dairy, as it may curdle.
Season and serve: Season the soup with salt and pepper to taste. Ladle the soup into bowls and garnish with fresh, chopped parsley.