Preheat the oven to 375 degrees F.
Spread the flour on a deep plate or in a shallow bowl and season generously with salt and pepper. Dredge the tilefish fillets in the flour.
Heat a large skillet over medium-high heat. Add 1 tablespoon each of the butter and olive oil. Add the fillets and sear until browned on one side, 2 to 3 minutes. At the same time, add the prosciutto and cook, stirring, until browned. Flip the fillets and place the skillet in the oven. Roast until the fish is just cooked through, about 10 minutes. Set the fillets on serving plates and drain the prosciutto on a paper towel-lined plate.
Return the skillet to the stovetop and heat the remaining 2 tablespoons butter and 1 tablespoon olive oil over medium heat. Add the white wine, parsley, capers and lemon juice. Bring to a boil and cook until reduced and thickened. Season with salt and pepper.
Pour the sauce over the fish, top with the prosciutto, and garnish with parsley sprigs. Serve immediately.