Pat both fish fillets dry with a paper towel. Season both sides of each fillet with ground black pepper. Wrap each fillet with thinly sliced prosciutto, so that one slice wraps around the top half and one slice wraps around the bottom half and the two slightly overlap in the middle.
Heat the olive oil in a large cast iron skillet over medium-high heat. Once the oil is hot, add the fish fillets and brown each side slightly, just about 2 to 3 minutes on each side. Be sure to watch the fillets carefully so that the prosciutto doesn't burn. Transfer the skillet to the middle rack of the oven and bake for about 6 to 7 minutes, until a thermometer inserted into the center of the fish reads 135°F. Use oven mitts or pot holders to remove the hot skillet from the oven and place onto your stove. Then transfer the fish to a plate and set it aside.
To the hot skillet (remember: the handle will still be very hot, so wear oven mitts for safety!), add the butter and allow it to melt down over medium-high heat. Just before the butter is completely melted, reduce the heat to medium and add the capers, garlic, parsley, lemon juice. Stir the ingredients together with a wooden spoon or heatproof silicon spatula and cook for 1 to 2 minutes, just until the garlic becomes soft and fragrant.
Set each fish fillet onto its own plate. Spoon the caper brown butter sauce over each fillet. Serve immediately.
Notes
Fish substitute: You can use cod instead of halibut.