Melt the butter in a large pot over medium heat. Add the onion, celery, and carrot. Sweat the veggies until softened, about 10 minutes, stirring frequently.
Add the garlic to the pot and cook it for about 1 minute, until very fragrant.
Begin sprinkling the flour into the pot, a bit at a time, stirring between additions so the flour coats the veggies. Cook the veggies and flour for about a minute, stirring frequently.
Begin adding broth to the pot, a bit at a time, whisking each addition in to eliminate any lumps formed by the flour.
Once all the broth has been added, stir in the milk, potatoes, bay leaf and thyme. Raise the heat and bring the liquid to a boil.
Lower the heat and allow the soup to simmer, stirring occasionally, until the base thickens a bit and the potatoes are tender, about 25 to 30 minutes.
Remove the pot from heat. Remove and discard the bay leaves. Stir in the vinegar and season the soup with salt and pepper to taste. Ladle into bowls and serve.