Pat fish filets dry with paper towels, then dredge fillets in flour. Shake off excess flour and set aside.
Melt 4 tablespoons of butter in 12 inch skillet over medium-high heat. When butter begins to foam, add fillets and cook until golden brown on the outside and thickest part of the fillet is opaque white throughout – about 3 to 5 minutes per side, depending on thickness. (If all the fillets do not fit in the pan it may be necessary to add an additional Tablespoon of Butter between batches.) Transfer fillets to serving plate and keep warm by tenting in foil or placing in a warm oven.
Deglaze the pan by adding vermouth and stirring and scrapping browned bits loose from the pan. Bring to a boil and reduce the volume by half.
Remove pan from heat and stir in 3 tablespoons of butter and fresh tarragon. Pour resulting sauce over fish and serve immediately.