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In a large bowl, whisk together 2 tablespoons of the oil, the warm water, and 1/4 teaspoon salt, then stream in the chickpea flour while whisking to avoid lumps. Let rest for 30 minutes.
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In a large skillet, heat 1 tablespoon of the olive oil over medium-high heat until hot but not smoking. Add the onion, 1/2 teaspoon salt, and pepper to taste and sauté, stirring frequently, until soft and brown, about 20 minutes.
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Set the oven on broil. Heat an 8-inch cast-iron skillet until smoking.
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Add 11/2 tablespoons olive oil, then immediately add 1/3 cup batter to the pan, swirling it around to evenly coat the bottom. Top with 1/4 of the caramelized onions and the 1/4 herbs and broil about 8 inches below the heat until lightly browned, 3 to 4 minutes. Transfer to a platter or cutting board, slice into wedges, season with salt, and serve immediately.