Position one oven rack in the middle setting. Position another rack in the lowest setting, and place a rimless baking sheet on the bottom rack. Preheat oven to 500°.
Melt butter in a large nonstick skillet over medium-high heat; add 1 teaspoon olive oil to pan. Add mushrooms to pan; sauté 10 minutes or until moisture evaporates. Sprinkle with salt and pepper; stir to combine.
Remove plastic wrap from Basic Pizza Dough; discard. Brush remaining 1 teaspoon oil evenly over dough. Remove preheated baking sheet from oven; close oven door. Slide dough onto preheated baking sheet, using a spatula as a guide. Bake on lowest oven rack at 500° for 8 minutes. Remove from oven.
Spread mushroom mixture in an even layer over crust, leaving a 1/4-inch border. Top with cheese. Slice prosciutto crosswise into thin strips; sprinkle over pizza. Bake on middle rack an additional 10 minutes or until crust is golden brown and cheese melts. Sprinkle with fresh parsley. Cut into 12 wedges.