In a large pot or Dutch oven, cook the Italian sausage over medium heat, breaking it up with a spoon until browned and cooked through, about 6-8 minutes.
Add the diced onion to the pot and cook until softened, about 4 minutes.
Stir in the minced garlic and cook for 1 minute until fragrant.
Sprinkle the flour over the sausage mixture and stir well to combine. Cook for 2 minutes to remove the raw flour taste.
Slowly pour in the chicken broth while stirring to avoid lumps.
Add the diced potatoes, dried thyme, salt, and black pepper. Bring the soup to a boil, then reduce heat to low and simmer until potatoes are tender, about 15-20 minutes.
In a separate small saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1 minute. Gradually whisk in the milk and cook, stirring constantly, until the mixture thickens, about 3-5 minutes.
Stir the milk mixture into the soup pot and cook for an additional 5 minutes until heated through and slightly thickened.
Remove the soup from heat and stir in the shredded cheddar cheese and sour cream until melted and smooth.
Taste and adjust seasoning with additional salt and pepper if needed.
Serve hot, garnished with chopped fresh parsley.