Whisk the chickpea flour, rosemary, salt, and pepper in a medium bowl. Gradually whisk in the water and a tablespoon of the olive oil until no lumps remain. Let the batter sit at room temperature for at least 30 minutes to allow the flour to fully hydrate.

Heat the topping olive oil in a 12-inch nonstick skillet over medium-high heat until shimmering. Add the onions, brown sugar, salt, and pepper and cook, stirring occasionally, until the onions are deeply browned and softened, 25 to 30 minutes. Stir in the vinegar, then transfer the onions to a bowl and wipe out the skillet.

Adjust an oven rack 4 inches from the broiler element and heat the broiler.

Heat a teaspoon of the remaining oil in a 10-inch oven-safe nonstick skillet over medium-high heat. Pour a portion of the batter into the skillet and swirl to coat the bottom evenly. Sprinkle a tablespoon of the Parmesan and a fifth of the onion mixture over the top.

Place the skillet under the broiler and cook until the edges are crisp and the top is blistered and browned, 2 to 3 minutes. Slide the flatbread onto a cutting board and repeat the process with the remaining oil, batter, cheese, and onions.

Serve hot, cut into wedges, while the edges are still crackling.