This recipe is inspired by the one in "Julia and Jacques Cooking at Home" (copyright Julia Child and Jacques Pepin, published by Alfred A. Knopf). Ask your fishmonger to clean the fish for you.
Sole: Heat a saute pan over medium-high. Season both sides of fish with salt and pepper. Place flour in a shallow dish. Dredge each fish in flour, turning and pressing lightly to coat. Shake off excess and place on a platter.
Swirl half of oil and butter in pan and, when foam subsides, place 1 fish in pan, white skin-side down. Saute until browned on first side, 4 to 5 minutes. Turn fish over carefully and saute until other side is crisp and golden brown, 4 to 5 minutes more. Transfer to a serving plate.
Repeat with remaining oil and butter to saute remaining fish, then transfer it to another plate.
Beurre noisette: Sprinkle half of parsley on each fish. Heat butter in a medium saucepan over high, swirling to melt. Cook until starting to brown, then remove from heat, and, as butter darkens to a hazelnut color, toss in capers and lemon juice; swirl to combine. Pour beurre noisette over fish and serve.