Peel and chop the carrots.
Saute carrots: Melt 1 tablespoon butter in a soup pan and cook the carrots and a pinch of salt for about 5 minutes, stirring several times in between. The carrots should be glossy and slightly softer.
Simmer: Add the vegetable broth, cover, bring to a boil, and cook on medium-low heat for about 15-20 minutes or until the carrots are soft.
Strain the soup and reserve the liquid. Puree the carrots with an immersion blender. Add a little of the cooking liquid when you blend the soup; it makes blending easier. Melt the remaining butter in the pot.
Make the roux: Melt 1 tablespoon of butter in the empty pot. Sprinkle the flour on top and stir well, do not let it get colored.
Add liquid: Slowly start adding the reserved cooking liquid while whisking continuously to avoid the formation of clumps. If that is still happening, you can blend the soup again after you return the blended carrots to the pot.
Add the milk and the pureed carrots to the pot, stir well and let bubble for another couple of minutes. If you like a thinner soup, you can add a bit more milk or broth.
Adjust the taste with salt and pepper. Add 1 teaspoon white balsamic vinegar or more to taste.
Serve sprinkled with parsley/ thyme/lovage/cayenne pepper and buttered toast, for instance.