Heat the oven to 200°F and set a wire rack inside a rimmed baking sheet. Whisk the chickpea flour, rosemary, salt, and pepper in a bowl, then slowly whisk in the water and oil until the batter is smooth and no lumps remain.

Heat two teaspoons of oil in an 8-inch nonstick skillet over medium-high heat—the oil should shimmer and move easily. Pour in a half cup of the batter and tilt the pan to coat the bottom evenly.

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, sprinkle a tablespoon of Parmesan over the top, and continue to cook until the second side is browned, 2 to 3 minutes.

Transfer the finished socca to the wire rack in the oven to keep warm and repeat the process with the remaining oil, batter, and cheese.

Heat two tablespoons of oil in a 12-inch nonstick skillet over high heat until shimmering. Stir in the brown sugar, salt, and onions, cooking until the onions begin to soften and release moisture, about 5 minutes.

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

Remove the pan from the heat and stir in the vinegar, then season with extra salt and pepper to taste. Top each cooked socca with a portion of the onion mixture, slice into wedges, and serve warm.