In a medium saucepan, heat over medium-high heat. Add the ground beef and cook for about 5-7 minutes, breaking it apart with a spatula until evenly browned and no longer pink. Drain any excess fat.

In the same saucepan, add two tablespoons of butter along with the chopped onion, diced carrots, and celery. Sauté for about 10 minutes until softened. Add minced garlic and cook for another minute.

Pour in the chicken broth along with the diced potatoes and the browned beef. Stir well to combine, then bring to a boil. Reduce heat to low, cover, and simmer for 10-12 minutes until the potatoes are tender.

In a separate skillet, melt 2 tablespoons of butter over medium heat and whisk in ¼ cup of flour to create a roux. Cook for about 3-5 minutes until it turns light golden brown.

Gradually add the cooked roux to the simmering soup, whisking constantly to avoid lumps. Allow the soup to return to a gentle boil.

Reduce the heat to low again, then fold in 8 ounces of Velveeta cheese and 1 cup of milk into the soup. Stir until cheese has melted completely.

Remove the pot from heat and gently fold in ½ cup of sour cream. Stir thoroughly until fully blended, then taste and season with salt and pepper as needed.