Caramelized Mushroom Flatbread Pizza

If using refrigerated pizza dough, make sure it sits out at room temperature for about an hour before beginning. Arrange a rack in the bottom third of the oven. Place a heavy, rimmed baking sheet upside down on the rack, and heat the oven to 450°F. Meanwhile, cut the mushrooms into 1/4-inch thick slices. Mince the 4 garlic cloves. Strip the leaves from the thyme sprigs until you have 2 teaspoons, then add to the garlic. Shred the mozzarella cheese until you get 1 cup.

Heat 1 tablespoon olive oil in a large skillet over medium heat until shimmering. Add half the mushrooms in a single layer and cook undisturbed until browned on the bottom, about 5 minutes. Season with 1/4 teaspoon of the salt and 1/8 teaspoon of the pepper and cook, stirring once or twice, until the mushrooms are browned all over and tender, 2 to 3 minutes more. Transfer to a medium bowl.

Add 1 tablespoon olive oil to the pan and repeat with the remaining mushrooms, salt, and pepper. About a minute before the mushrooms are ready, add the garlic and thyme and sauté until fragrant, about 1 minute. Transfer to the bowl of mushrooms and stir to combine. Set aside to cool while you prepare the dough.

Divide the pizza dough into 2 pieces. If the dough is sticky, dust with a little bit of all-purpose flour. Sprinkle a 12-inch piece of parchment paper with cornmeal and place one of the pieces of dough on it. Use the heel of your hand to press the dough flat. Using your hands or a rolling pin, work from the middle out to shape the dough into a 10-inch round. The dough will stick to the parchment; if it starts to shrink back, wait a few minutes to let it relax and continue shaping.

Brush the dough with 1 tablespoon olive oil, then scatter half of the mushroom mixture onto the dough, leaving a 1/2-inch border. Sprinkle with half of the mozzarella cheese. Carefully remove the preheated baking sheet from the oven and use the parchment paper to slide the flatbread directly onto the back of the baking sheet.

Bake until the edges are golden-brown and crisp and the cheese is melted,10 to 12 minutes. Meanwhile, grate enough Parmesan cheese until you have 2 tablespoons. Chop until you have 2 tablespoons chopped parsley leaves.

Remove the baking sheet from the oven and carefully transfer the flatbread to a clean cutting board. Save the parchment paper to make a second flatbread and return the baking sheet to the oven. Top the baked flatbread with half of the Parmesan and half of the parsley. Cool for 2 minutes, then slice and serve. Repeat with the remaining dough, olive oil, and toppings.