In a large pot or Dutch oven, melt the butter over medium heat.
Add the sliced smoked sausage and cook until browned, about 5 minutes. Remove the sausage with a slotted spoon and set aside.
In the same pot, add the chopped onion, diced celery, and minced garlic. Cook until the vegetables are softened, about 5 minutes.
Sprinkle the flour over the vegetables and stir well to coat. Cook the flour for 2 minutes, stirring constantly to remove the raw flour taste.
Gradually whisk in the chicken broth, making sure to dissolve any lumps.
Add the diced potatoes, dried thyme, smoked paprika, salt, and pepper. Bring the mixture to a boil, then reduce heat and simmer until the potatoes are tender, about 15 minutes.
Once the potatoes are tender, stir in the milk and heavy cream.
Return the browned sausage to the pot and cook for an additional 5 minutes, allowing the flavors to meld and the soup to heat through. Do not boil after adding milk and cream.
Taste and adjust seasoning with salt and pepper as needed.
Ladle the chowder into bowls and garnish with chopped fresh parsley before serving.