In a large Dutch oven over medium-high heat, add the ground beef. Season lightly with salt and pepper and cook until browned and crumbly, about 5–7 minutes. Drain excess fat, leaving a tablespoon for flavor.

Add butter to the pot. When it melts and begins to bubble, add the chopped onion, carrots, and celery. Sauté until the onion is translucent and sweet, about 4–5 minutes. Stir in the garlic for the last 30 seconds until fragrant.

Sprinkle the flour over the vegetable and beef mixture and stir for about 1 minute to cook the raw flour taste away.

Gradually pour in the beef broth while stirring to avoid lumps. Add the diced potatoes and smoked paprika. Bring the soup to a gentle boil, then reduce heat and simmer until the potatoes are tender, about 12–15 minutes.

Lower the heat to low. Stir in the milk or half-and-half, then slowly add 1 cup of shredded cheddar, stirring until creamy and melted.

If you want a thicker soup, whisk an extra tablespoon of flour with a little cold milk and stir in, cooking for another 2 minutes. Taste and adjust seasoning with salt and pepper, then fold in the remaining cheese for extra cheesiness.

Ladle into bowls and garnish with chives, a sprinkle of black pepper, and optional bacon bits or pickles for that cheeseburger-inspired twist.