Rinse fillets and pat dry. In small bowl combine flour, salt, pepper and thyme. Pour the dry mixture out onto a plate and dredge each fillet, lightly covering all surfaces; set aside. Cut stick butter into 1 oz. pats; set aside.
Heat 1/3 of oil in large skillet over medium-high heat. Place 2 fillets (or more, just don't crowd skillet surface) in skillet and saute for several minutes on each side. Remove fillets from skillet, place on plate, and sprinkle with 1/3 parsley. Set in your oven as the remaining heat of fillet will crisp the fresh parsley. Repeat until all fillets are cooked.
Pour off the oil and wipe skillet dry. Return skillet to heat and begin melting butter, 1 oz at a time until bubbling hot, but not brown. Quickly add the lemon juice and slices (capers if using); stir gently in the skillet until combined and warm. Pour over fillets and serve.