In a large pan over medium-high heat, heat 1 tablespoon of the butter. Add the ground beef, finely diced onion, finely diced carrots, dried basil, and dried parsley. Cook, breaking up the beef with a spoon, until the beef is browned and the vegetables are tender.
Drain off any excess grease from the pan and return the meat-vegetable mixture to the pan.
Add both cans of chicken broth (14.5 oz each) and the diced, peeled Yukon Gold potatoes to the pan. Bring the mixture to a boil, then reduce the heat to low, cover, and simmer for 10–12 minutes or until the potatoes are tender.
While the potatoes are cooking, melt the remaining butter in a small saucepan over medium heat. Add the all-purpose flour and whisk continuously to form a smooth paste.
Slowly whisk the milk into the flour-butter paste until the mixture is smooth and slightly thickened.
When the potatoes are tender, stir the milk mixture into the soup until fully incorporated.
Add the Velveeta cheese, salt, and pepper to the soup. Cook over low heat, stirring frequently, until the cheese is completely melted and the soup is smooth.
Remove the soup from the heat and stir in the sour cream until combined. Serve immediately.