To prepare crust, combine 5 tablespoons water, 1 teaspoon oil, and yeast in a bowl. Let stand 2 minutes. Weigh or lightly spoon 5 ounces bread flour into dry measuring cups; level with a knife. Sprinkle bread flour over yeast mixture. Add semolina flour, rosemary, and salt. Stir until just combined. Turn dough onto counter; knead 1 minute. Place dough in a medium bowl coated with cooking spray, turning to coat top. Cover and place in a warm place, free from drafts, 40 minutes.
Position an oven rack in lowest setting. Place a pizza stone on lowest rack. Preheat oven and pizza stone to 500°; leave stone in oven 30 minutes.
To prepare topping, heat a nonstick skillet over medium-high heat. Add 2 teaspoons oil; swirl. Add mushrooms; sprinkle with 1/4 teaspoon salt. Sauté for 2 minutes. Add garlic and pepper; sauté 3 minutes.
Turn dough out onto a lightly floured surface. Roll dough out to a thin 14-inch circle. Transfer dough to a baking sheet dusted with cornmeal. Brush the dough with 1 teaspoon oil. Slide dough onto the preheated pizza stone, using a spatula as a guide. Reduce oven temperature to 475°; bake 5 minutes. Remove from oven. Spread sauce over crust, leaving a 1/2-inch border; top with mozzarella cheese, mushroom mixture, and Parmigiano. Bake 4 minutes or until crust is browned. Slice pizza into 8 wedges.