Mushroom, Caramelized Onion, and Goat Cheese Pizza Flatbread

Name: Mushroom, Caramelized Onion, and Goat Cheese Pizza Flatbread
Prep time: 40 min
Cook time: 1 h
Total time: 1 h 40 min
Cuisine: American
Ingredients: 2 Tbsp. olive oil, divided; 1 large sweet onion, sliced; ¾ tsp. salt, divided, or more to taste; 8 oz. sliced brown mushrooms; ¼ tsp. pepper, or more to taste; 1 ¼ lb. prepared pizza dough, homemade or store-bought; ½ cup crumbled goat cheese; 1 Tbsp. fresh thyme leaves (optional)