Adjust an oven rack to the lowest position, place a pizza stone or inverted baking sheet on the rack, and heat the oven to 500°F.

Heat the olive oil in a 12-inch nonstick skillet over medium heat until shimmering. Add the sliced portobello mushrooms, cover, and cook until they are wilted and have released their liquid, about 5 minutes.

Uncover the skillet and continue to cook until the liquid has evaporated and the mushrooms are dry and well-browned, 5 to 10 minutes. Stir in the garlic and cook until fragrant, then season with salt and pepper and transfer to a bowl.

Divide the dough into two pieces and stretch each into a 12-inch round on a piece of parchment paper. Spread the pizza sauce thinly over each round, leaving a 1/2-inch border at the edges.

Distribute the spinach, mozzarella, goat cheese, Parmesan, and the mushroom mixture evenly over the two pizzas. The spinach should be scattered loosely so it heats through properly.

Slide the pizzas (with the parchment) onto the hot stone or baking sheet and bake until the crust is well-browned and the cheese is bubbly, 10 to 12 minutes.

Serve hot, while the goat cheese is soft and the crust is still crisp.