Brown the Ground Beef: In a large soup pot or Dutch oven, brown ground beef over medium heat until fully cooked. Drain excess fat and set aside.
Sauté Aromatics: In the same pot, melt butter. Add diced onions, garlic, and grated carrot. Cook until softened, about 3–4 minutes.
Add Flour to Make Roux: Sprinkle in flour, stir constantly for 1–2 minutes to create a smooth roux.
Pour in Broth & Simmer: Gradually whisk in beef broth. Add diced potatoes and bring to a simmer. Cover and cook for 10–12 minutes or until potatoes are fork-tender.
Reintroduce the Beef: Return the cooked ground beef to the pot and stir to combine.
Finish with Dairy: Reduce heat to low. Stir in milk, heavy cream, and shredded cheddar cheese. Stir gently until cheese is melted and soup is velvety.
Taste & Serve: Adjust seasoning with salt and pepper. Ladle into bowls and top with your favorite cheeseburger toppings.