Clean the mushrooms by brushing the tops with a clean sponge. Don't wash them! Remove and discard the stems and slice the caps 1/4 to 1/2 inch thick. In a large saute pan, heat the butter and 2 tablespoons of the olive oil until bubbly. Add the mushrooms, salt, and pepper to the pan, and saute for 3 minutes over medium heat, tossing frequently. Reduce the heat to low and saute for another 2 to 3 minutes, until cooked through.
Meanwhile, arrange the arugula on 4 lunch plates (or on a platter) and cover each portion with 2 slices of prosciutto. When the mushrooms are cooked, add the sherry vinegar and the remaining 2 tablespoons of olive oil to the hot pan. Spoon the mushrooms and sauce on top of the prosciutto. With a vegetable peeler, make large shavings of Parmesan cheese and place on top of the hot mushrooms. Sprinkle with the sun-dried tomatoes, parsley leaves, salt, and pepper, and serve warm.
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