Preheat the oven to 375 degrees F.

Put the flour on a deep plate or in a shallow bowl and season well with salt and pepper. Dredge the fish in the flour. Put a large skillet over medium-high heat, add 1 tablespoon oil and the butter and get the skillet hot. Add the fillets and cook until browned on 1 side, 2 to 3 minutes. At the same time, add the prosciutto and cook, stirring, to brown. Then flip the fish, put the skillet in the oven, and roast until the fish is just cooked through, about 10 minutes.

Remove the fish to 2 serving plates. Dump the prosciutto out onto paper towels to drain. Put the skillet back over medium heat. Add another tablespoon olive oil, the white wine, lemon juice, capers, the remaining 2 tablespoons butter and the parsley and bring to a boil; boil until reduced and thickened. Season with salt and pepper. Pour the sauce over the fish, top with the prosciutto, and serve immediately.