Place a rack in the lowest position and heat oven to 500F. Let pizza dough sit at room temperature for 30 minutes if cold. Dust a large rimmed baking sheet with cornmeal.

Heat 1 tablespoon olive oil in a large skillet over medium-high heat until shimmering. Add mushrooms in a single layer and cook undisturbed for 4 minutes until deeply browned on the bottom. Stir and cook 3 minutes more until tender and most moisture has evaporated.

Add garlic and red pepper flakes if using. Cook 30 seconds until fragrant. Add spinach by handfuls, tossing until wilted, about 2 minutes total. Season with 1/4 teaspoon salt. Transfer to a plate to cool slightly.

Stretch dough into a 12x16 inch rectangle on the prepared sheet. Brush with remaining 1 tablespoon olive oil. Scatter mozzarella evenly, leaving a 1/2 inch border. Distribute mushroom-spinach mixture over cheese, then sprinkle with Parmesan.

Bake on the lowest rack for 12-14 minutes until the crust is deep golden and crisp on the bottom, and cheese is bubbling with browned spots. Let cool 3 minutes, then finish with flaky salt and slice into squares.