Heat 2 Tablespoons butter in a large, deep skillet. Add the chicken, skin side down, & cook until lightly brown, about 5 minutes. Turn the pieces over & season with Essence, salt & pepper to taste. Add the wine to the skillet, cover, & simmer for 20 minutes longer.
While the chicken is cooking, melt the remaining 2 Tablespoons butter in a 1- quart pan. Add the flour & stir to blend. When the mixture starts to bubble, pour in the milk, whisking to eliminate any lumps. Bring to a boil & simmer for 5 minutes. Remove pan from the heat, stir in the mustard, cream, & tarragon. Season well with Essence.
When the chicken is done, remove it & the mushrooms with a slotted spoon to a serving platter. Cover & keep warm. Spoon off & discard any fat that remains in the skillet, then boil the liquid until about 1/4 cup remains.
Stir the reduced liquid into the sauce & reheat. Season to taste with salt & pepper & serve over chicken & mushrooms. Serves 3 - 4