Melt butter in a large pot or Dutch oven over medium heat. Add onion, carrots, and celery, cooking until softened, about 5-7 minutes.
Stir in garlic and cook for another minute until fragrant.
Sprinkle flour over the vegetables and stir well, cooking for 2 minutes to remove the raw flour taste.
Gradually pour in the broth while stirring constantly to prevent lumps from forming.
Add the milk, potatoes, thyme, and bay leaf. Bring to a boil, then reduce heat and simmer for 15-20 minutes until potatoes are tender.
Stir in the turkey and cook for another 5 minutes until heated through.
Remove bay leaf, season with salt and pepper to taste, and garnish with fresh parsley before serving.