Preheat oven to 350°F.
Line 9 x 13-inch baking dish with foil.
Melt butter in medium skillet over medium-high heat.
Add mushrooms and onion and saute until softened, about 5 minutes.
Arrange fillets in dish.
Pour vermouth over fish.
Dot each piece generously with butter and sprinkle with tarragon, salt and pepper to taste.
Spoon mushroom mixture evenly over top.
Cover with foil and bake until fish loses its translucency, about 30 minutes.