Steam asparagus spears until tender.
Place in the bowl of a food processor and pulse until a nice purée is formed, scraping the sides of the bowl as needed. Set aside for later use.
In a small saucepan combine wine, half of the tarragon with the freshly ground salt and pepper and bring to a boil. Reduce heat and simmer uncovered until wine is reduced to 1/4 cup, about 20 minutes. Remove from heat, strain and set aside.
Melt butter in a large Dutch oven. Add flour stirring constantly and cook over medium heat for about 1 minute to obtain a "roux", a thick paste.
Gradually add chicken or vegetable broth, stirring constantly until well blended.
Add asparagus purée, reduced wine mixture, milk and remaining tarragon and bring to a boil.
Reduce heat and simmer uncovered for another 20 minutes.
Serve hot with fresh tarragon sprigs for garnish.