Brown the beef:
In a large pot or Dutch oven, cook the ground beef over medium heat until browned. Drain excess fat and set aside.
Sauté the vegetables:
In the same pot, melt 2 tablespoons of butter. Add diced onion, celery, and garlic; cook until fragrant and softened, about 5 minutes.
Add broth and potatoes:
Stir in the chicken broth, shredded carrots, and diced potatoes. Bring to a boil, then reduce to a simmer. Cook for 10–12 minutes, until the potatoes are tender.
Make it creamy:
In a small saucepan, melt the remaining 2 tablespoons of butter. Whisk in the flour and cook for 1 minute. Slowly whisk in the milk until smooth and thickened. Pour this mixture into the soup pot.
Add cheese and beef:
Return the cooked beef to the pot, then stir in shredded cheddar cheese. Mix until melted and creamy. Season with paprika, salt, and pepper to taste.
Finish and serve:
For extra richness, stir in sour cream before serving. Ladle the creamy cheeseburger soup into bowls and top with green onions, bacon bits, or extra cheese.