In a large pot or Dutch oven, cook the Italian sausage over medium heat, breaking it apart with a spoon, until browned and cooked through, about 7 minutes.
Remove the sausage from the pot and set aside, leaving the rendered fat in the pot.
Add the diced onion to the pot and sauté in the sausage fat over medium heat until softened, about 5 minutes.
Add the minced garlic and cook for 1 minute until fragrant.
Stir in the butter and let it melt, then sprinkle the flour over the onions and garlic. Stir continuously for 2 minutes to form a roux.
Slowly whisk in the chicken broth, making sure to dissolve any lumps from the flour.
Add the diced potatoes, dried thyme, smoked paprika, salt, and black pepper to the pot. Bring the mixture to a boil, then reduce heat to a simmer.
Cover and cook until the potatoes are tender, about 15 minutes.
Return the cooked sausage to the pot and stir in the whole milk. Heat through for 5 minutes without boiling.
Adjust seasoning with additional salt and pepper if needed.
Remove from heat and stir in the chopped fresh parsley.
Ladle the soup into bowls and serve hot.