Dust the counter surface with bench flour. Stretch out the dough to a 12-inch circle. Drizzle the hot skillet lightly with just 1 tablespoon of oil. Lift up the skillet by the handle and swirl to coat the bottom of the pan completely.
Add the dough to the hot skillet and carefully stretch it to go halfway up the sides. Press your fingers into the dough to make small indents. Sprinkle with the remaining 2 tablespoons olive oil, the red onions and rosemary. Top with the freshly grated Parmesan.
Bake on the middle rack until lightly golden brown, about 35 minutes. Slice into wedges to serve.