Heat olive oil in a large pot over medium heat. Add sausage and cook until browned, breaking it up as it cooks.
Add onion, carrots, and celery. Sauté for 5-6 minutes until vegetables begin to soften.
Stir in garlic and cook for another minute until fragrant.
Sprinkle flour over the mixture and stir well to combine. Cook for 1-2 minutes to eliminate the raw flour taste.
Gradually add chicken broth while stirring, then add milk and cream.
Add diced potatoes and thyme. Bring to a boil, then reduce heat and simmer for 20-25 minutes, or until potatoes are tender.
Season with salt and pepper to taste.
Ladle into bowls and garnish with chopped parsley if desired. Serve hot.