Preheat oven to 450 degrees. Place a rimmed baking sheet upside-down in oven and heat 5 minutes.
Meanwhile, stretch out dough on a lightly floured piece of parchment into a 12-by-16-inch rectangle, about 1/8 inch thick. Drizzle with oil and, using parchment, transfer to prepared baking sheet. Bake until puffed and golden, 8 to 10 minutes.
Scatter sausage and broccoli over crust; top with 1 cup pepperoncini sauce. Bake until toppings are heated through, about 5 minutes. Remove from oven; top with Parmesan. Transfer to a cutting board, slice, and serve with pepper flakes.