Mash 4 tablespoons softened butter and mustard until smooth. Chill.
Season fish on both sides with salt.
Preheat oven to 400 degrees F.
In a small saucepan, combine vermouth and shallot. Over high heat, bring to a boil and reduce to about 1/4 cup, about 5 minutes. Turn the heat down to very low and add the mustard butter, a tablespoon at a time, whisking after each addition until butter is incorporated. Keep sauce warm.
Heat an oven-proof pan to medium-high. Add 1 tablespoon butter and olive oil. Place fish in pan and cook until nicely browned on the bottom, about 1 1/2 to 2 minutes. Carefully flip fish and place pan in the oven. Roast the fish for about 10 minutes, or until done.
Stir the chopped tarragon into the sauce. Spoon sauce over warm plate. Lay fish on top of sauce. Season with pepper. Serve with lemon wedges.