Adjust an oven rack to the middle position and heat the oven to 200°F. Set a wire rack inside a rimmed baking sheet and place it in the oven to keep the finished pancakes warm.

Whisk the chickpea flour, rosemary, salt, and pepper together in a bowl. Slowly whisk in the water and oil until the mixture is combined and completely smooth.

Heat oil in an 8-inch nonstick skillet over medium-high heat until it shimmers. Pour in some batter and tilt the pan to coat the bottom evenly, then reduce the heat to medium and cook until the edges are crisp and the bottom is golden brown, 3 to 5 minutes.

Flip the socca and sprinkle Parmesan over the top. Continue to cook until the second side is browned, about 2 to 3 minutes, then transfer the pancake to the wire rack in the oven and repeat with the remaining oil, batter, and cheese.

Heat oil in a 12-inch nonstick skillet over high heat until shimmering. Stir in the brown sugar, salt, and onions to coat, then cook for about 5 minutes until the onions begin to soften and release moisture.

Reduce the heat to medium and cook, stirring often, for 30 to 35 minutes until the onions are deeply caramelized. If they begin to scorch, lower the heat; if they aren't browning after 20 minutes, increase the heat slightly.

Stir in the vinegar after removing from the heat and season with additional salt and pepper. Top each warm socca with the onion mixture, slice into wedges, and serve.