Mince carrots, onions, and celery in a food processor and saute in heavy pot with oil. Mince mushrooms in a food processor, add to pot along with tarragon, salt, and pepper. Cook for about 15 minutes. Do not burn.
Add stock and cream and bring to a boil. Mix melted butter and flour until smooth and whip into soup. Bring to a boil, stirring constantly until all of the flour and butter mixture is incorporated and the soup is thickened. Simmer for 30 minutes and serve. Adjust seasonings with salt and pepper, to taste.
Recipe courtesy of The Terrace Restaurant, Kennett Square