1. 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.

2. 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.

3. Repeat with remaining oil and butter to saute remaining fish, then transfer it to another plate.

4. 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.