Step 1Heat the oven to 400 degrees. Melt the butter over medium heat in a large (13-inch) ovenproof skillet. Mince 2 slices of prosciutto and add to the butter. Cook until prosciutto starts to render its fat, about 2 to 3 minutes. Add mushrooms and sprinkle with one-eighth teaspoon salt.
Step 2Cook, stirring occasionally, until the mushrooms begin to give off their moisture, about 5 minutes. Add the garlic and parsley, and cook until the garlic is fragrant, about 1 1/2 to 2 minutes. Transfer the mixture to a bowl and cool slightly.
Step 3Season the fish inside and out with a very small amount (one-fourth teaspoon) of salt. Spoon about 3 tablespoons of the cooked mushrooms into the belly cavity of the fish, reserving the rest. Wrap the fish snugly in the remaining prosciutto slices, leaving the head exposed and with the middle of the slices sealing the belly. It will take 6 or 7 slices to enclose the fish.
Step 4Wind a length of butcher's twine around the fish to hold the prosciutto in place snugly. Inevitably, one side of the fish will have a more uniform wrapping of prosciutto than the other. This will be the presentation side, so knot string on the other side.
Step 5Clean the mushroom pan and return it to high heat. Add oil and when it is nearly smoking, place the fish in the pan, presentation-side down. Depending on the size of the pan, you may need to arrange the fish to make sure all of the prosciutto wrapping comes in contact with the heat. Sear until the prosciutto has darkened and begun to crisp, 3 to 5 minutes.
Step 6Using a wide spatula, carefully turn the fish over so the presentation side is up and cook another minute to begin crisping the prosciutto. Scatter the remaining mushrooms on both sides of the fish and place the skillet in the oven. Cook until a small knife penetrates the flesh easily, about 10 to 15 minutes.
Step 7Carefully transfer the fish to a warm platter. Use scissors to cut the string and remove it. Let stand a few minutes while scattering the mushrooms over the fish and serve hot.