In a large pot or Dutch oven, heat a medium skillet over medium heat. Add the sliced smoked sausage and cook until browned on both sides, about 5 minutes. Remove the sausage from the skillet and set aside.
In the same large pot, melt the unsalted butter over medium heat. Add the chopped onion, diced celery, and diced carrots. Cook, stirring occasionally, until the vegetables are softened, about 5-7 minutes.
Add the minced garlic and cook for 1 minute until fragrant.
Sprinkle the all-purpose flour over the vegetables and stir well to coat. Cook for 2 minutes to remove the raw flour taste, stirring constantly.
Slowly pour in the chicken broth while stirring to avoid lumps. Add the diced potatoes, dried thyme, smoked paprika, salt, and pepper. Bring the mixture to a boil, then reduce heat to medium-low and simmer until the potatoes are tender, about 15 minutes.
Once the potatoes are tender, stir in the browned sausage, whole milk, and heavy cream. Heat the soup gently, but do not boil, until warmed through, about 5 minutes.
Taste and adjust seasoning with additional salt and pepper if needed.
Remove the pot from heat and stir in the chopped fresh parsley.
Ladle the chowder into bowls and serve hot.