Caramelized Mushroom Flatbread Pizza

Name: Caramelized Mushroom Flatbread Pizza
Ingredients: 1 pound pizza dough, at room temperature at least 1 hour; 1 pound cremini mushrooms; 4 cloves garlic; 2 teaspoons fresh thyme leaves; 1 cup shredded part-skim mozzarella cheese, divided; 4 tablespoons olive oil, divided; 1/2 teaspoon kosher salt, divided; 1/4 teaspoon freshly ground black pepper, divided; All-purpose flour, for dusting; Finely ground cornmeal, for dusting; 2 tablespoons grated Parmesan cheese, divided; 2 tablespoons chopped fresh parsley leaves, divided