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.
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.
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.
To store, wrap cooled focaccia in plastic and keep at room temperature, up to 2 days.