Heat a 3-quart saucepan over medium-high heat. Add beef; cook and stir until browned and crumbly, 5 to 7 minutes. Remove beef and set aside.

Heat 1 tablespoon butter in the same saucepan; add onion, carrots, celery, basil, and parsley. Cook and stir until vegetables are tender, about 3 to 5 minutes. Add broth, has brown potatoes, and browned beef; bring to a boil. Reduce heat, cover and simmer until potatoes are tender, about 5 to 10 minutes.

Meanwhile, melt remaining 3 tablespoons butter in a small skillet. Add flour; cook and stir until smooth, about 1 minute. Add to soup; cook and stir soup for 2 minutes. Reduce heat to low.

Add cheese, milk, salt, and pepper; cook and stir until cheese melts. Remove from heat; add sour cream and stir well. Serve immediately.