Preparation of Ground Beef: In a large skillet, sauté the ground beef over medium heat until it is thoroughly browned. Once cooked, drain off the excess fat and place the beef into a Crock Pot.
Combining Ingredients: Add the finely chopped onion, minced garlic, and diced potatoes to the Crock Pot along with the beef broth. Stir these components to integrate them fully.
Incorporation of Cream Elements: Introduce the cream of mushroom soup into the Crock Pot along with the milk or half-and-half. Season with dried thyme, dried parsley, and an appropriate amount of salt and pepper, mixing well to ensure all ingredients are evenly distributed.
Cooking: Cover the Crock Pot and set it to cook on low for 6 to 7 hours, or on high for 3 to 4 hours. The duration should be sufficient for the potatoes to soften and the flavors to meld, resulting in a slightly thickened soup.
Final Additions: Following the cooking period, incorporate the shredded cheddar cheese and sour cream into the soup if desired, stirring until the soup achieves a creamy consistency and the cheese is melted.
Serving: Ladle the soup into individual serving bowls, garnishing with freshly chopped parsley if preferred. Serve immediately while hot.