Cook the beef:
In a large saucepan over medium heat, cook and crumble the beef until no longer pink. Drain and set aside.
Sauté the vegetables:
In the same saucepan, sauté onion, carrots, celery, basil, and parsley in 1 tablespoon butter until vegetables are tender, about 10 minutes.
Add broth and potatoes:
Add chicken broth, potatoes, and beef to the saucepan. Bring to a boil.
Simmer:
Reduce heat and simmer, covered, until potatoes are tender, about 10-12 minutes.
Prepare the flour mixture:
In a small skillet, melt the remaining butter. Add flour and cook, stirring, until bubbly (about 3-5 minutes).
Thicken the soup:
Add the flour mixture to the soup and bring to a boil. Cook and stir for 2 minutes.
Add cheese and milk:
Reduce heat to low. Stir in shredded cheddar cheese, milk, salt, and pepper. Cook until the cheese melts.
Finish with sour cream:
Remove from heat and blend in the sour cream.
Serve:
Serve hot and enjoy your homemade cheeseburger soup!