1. Preheat oven to 425 degrees. Coat an 11-by-17-inch rimmed baking sheet with 2 tablespoons oil. Place dough on sheet, stretching to fit (if dough springs back, let rest 20 minutes and repeat). Drizzle edges with 1 tablespoon oil. Cover with plastic wrap and let sit 15 minutes.

2. Meanwhile, in a large skillet, heat 1 teaspoon oil over medium-high. Add onions and season with salt and pepper. Cook, stirring occasionally, until onions are tender and golden brown in spots, 12 minutes. Add vinegar and cook, stirring and scraping up browned bits with a wooden spoon, 1 minute.

3. Remove plastic from dough. With your fingertips, poke dimples all over dough, then top with cheese, browned onions, and red-pepper flakes. Bake until golden brown around edges, about 30 minutes. Let cool on sheet 5 minutes. Cut into 20 pieces and serve warm or at room temperature.