Preheat oven to 450ºF with rack in the lowest position. On a baking sheet lined with parchment paper, stretch out the dough to a 16-by-10-inch rectangle. Bake until lightly golden, about 7 minutes.
Meanwhile, heat 2 tablespoons oil in a large nonstick skillet over medium-high.
Add mushrooms and cook, stirring occasionally, until golden, about 12 minutes.
Remove from heat and stir in parsley and ½ teaspoon each salt and pepper.
Stir together ricotta, mozzarella, garlic, and remaining 2 tablespoons oil, ½ teaspoon salt, and ¼ teaspoon pepper in a medium bowl.
Spread over prebaked crust, leaving a 1-inch border. Arrange mushroom mixture on top.
Bake until golden, about 12 minutes. Cut into slices and serve.