Preheat the oven to 450-500°F, placing a pizza stone in if using. Allow pizza dough to come to room temperature.

Heat 2 tablespoons olive oil in a large skillet over medium to medium-high heat. Stir in mushrooms, shallots, chili flakes, garlic, thyme, and oregano. Cook until mushrooms are tender and browned, about 5-6 minutes. Season with salt and pepper, add a splash of wine or dry sherry, and cook until wine has evaporated and mushrooms are caramelized, about 3-5 minutes. Remove from heat and set aside to cool.

Dust a pizza peel with cornmeal or flour. Stretch or roll the dough into a 10-inch circle and place it onto the pizza pan, baking sheet, or pizza peel.

Brush the pizza with a thin layer of remaining olive oil. Top with ¾ of the mozzarella, then the mushroom mixture, and sprinkle remaining cheese over the mushrooms.

Transfer the pizza to the preheated pizza stone or baking sheet. Bake until the cheese and crust are browned and cheese is melted, about 8-10 minutes.

Once out of the oven, sprinkle with chopped parsley and lemon zest, slice into pieces, and serve immediately.