In a large pot over medium heat, melt 1 tablespoon of butter. Add ground beef, onion, carrots, and celery. Cook and stir until beef is browned and crumbly, about 5–7 minutes.

Drain excess fat, then stir in remaining 3 tablespoons of butter until melted. Sprinkle flour over the mixture and cook for 1–2 minutes, stirring constantly to form a roux.

Gradually whisk in chicken broth and milk, ensuring there are no lumps. Bring to a gentle boil and add cubed potatoes. Reduce heat to a simmer and cook until potatoes are tender, about 15–20 minutes.

Stir in shredded cheddar cheese until fully melted and creamy. Season with salt and pepper to taste.

Remove from heat and let the soup rest for 5 minutes before serving. Garnish with extra cheese or chopped green onions if desired.