Step 1Rinse the fish fillets under cold water and pat dry on paper towels. Season both sides of fish with salt and pepper to taste and set aside.
Step 2In a small nonstick skillet, reduce the sherry vinegar by half, about 30 seconds, and set aside.
Step 3In a heavy medium skillet, heat one-fourth cup of the butter over medium-low heat until melted, about 3 minutes. After the butter melts, begin whisking until it turns nut brown, about 4 to 5 minutes. Remove the butter from the heat and set it aside while preparing the fish.
Step 4In a large skillet, heat 1 tablespoon butter and the oil over medium heat until hot. Add the fish fillets presentation-side down in the hot skillet and brown 2 1/2 to 3 minutes on the first side, then turn and cook for 30 to 45 seconds. The fish should not be crowded in the pan; if necessary, cook the fillets in two batches.
Step 5Arrange the fish on serving plates. Sprinkle with parsley. Whisk reduced vinegar, capers and a pinch of salt and pepper into reserved brown butter and spoon over the fish.