Melt butter in Dutch oven.
Sprinkle chicken with salt and pepper. Brown on all sides about 10 minutes. Remove from pan and set aside.
Pour the Cognac into the pan and flame with a match. When flames subside, stir in flour and tarragon. Cook 2 minutes stirring constantly. Gradually whisk in 3 cups of Champagne. Place chicken back into pan, cover and simmer until tender, about 45 minutes.
Remove chicken and stir in the remaining Champagne and heavy cream. Cook over high heat until reduced and slightly thickened.
Pour over chicken and serve.