In a bowl, sprinkle yeast over 1 1/4 cups warm (110º) water. Let stand until softened, about 5 minutes. Add salt, onion, 1 1/2 tablespoons olive oil, and 2 1/2 cups flour.
If kneading by hand, stir vigorously until dough is stretchy, about 5 minutes. Stir in 1 1/4 cups flour. Scrape dough onto a lightly floured board and knead until smooth, elastic, and no longer sticky, 10 to 12 minutes, adding flour as required to prevent sticking.
If kneading with a dough hook, beat on low speed until flour is incorporated. Add 1 1/4 cups flour. Beat on low speed to blend, then on high speed until dough no longer feels sticky, and pulls cleanly from bowl, 10 to 12 minutes. If dough is still sticky, beat in flour, 1 tablespoon at a time.
Place dough in an oiled bowl, turn over, cover airtight, and let rise in a warm place until doubled, about 1 hour.
On a lightly floured board or with a dough hook, briefly knead dough to expel air. Cut into 6 equal pieces and shape each into a smooth ball.
Flatten 1 ball by hand, then roll into an 8-inch round (or into a 7-in. round if using frozen bread dough), reflouring board and rolling pin as needed to prevent sticking. Brush round with about 3/4 teaspoon olive oil and sprinkle with about 1 teaspoon rosemary, a scant 2 tablespoons parmesan, and pepper. Lightly press in seasonings. Cover a plate with plastic wrap. Place seasoned round on wrap, cover with plastic wrap, and chill. Repeat to shape remaining balls. As each round is finished, add to stack, cover with plastic wrap, and chill. (If making dough ahead, wrap stack airtight and chill up to 6 hours.)
Place a few dough rounds at a time, seasoned side up, on an oiled grill over a solid bed of medium-hot coals, or over medium-high heat on a gas grill (you can hold your hand at grill level only 3 to 4 seconds). Cover gas grill. With a wide spatula, occasionally lift and rotate breads for even browning. When bottoms are well browned, 2 to 3 minutes, turn breads over. Cook until seasoned sides are light brown, 1 to 2 more minutes.
Cut each flatbread into about 6 wedges. Dip into olive oil to eat.