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In a blender, combine 1 cup water with the olive oil, chickpea flour, salt, and pepper. Blend until smooth. (Alternatively, sift the chickpea flour, salt, and pepper into a bowl. Whisk in the water and olive oil until smooth and combined.) Cover and let the flatbread batter sit at room temperature for 30 to 60 minutes to thicken slightly.
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Heat the oven to 475°F. Place a 10-inch cast-iron skillet in the oven for 10 minutes. Carefully remove the skillet from the oven and swirl in the ghee. Pour the flatbread batter into the hot pan and, working fairly quickly, carefully layer the toppings over the batter in the following order: red onion, olives, goat cheese.
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Return the skillet to the oven and bake until the edges are deep golden brown, and the center is mostly set, 18 to 25 minutes. Remove from the oven and carefully remove the flatbread from the pan to a cutting board. The socca will continue to cook from residual heat.
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Toss the arugula with the lemon juice, a drizzle of olive oil, the pepper flakes, and salt and pepper to taste. Scatter over the socca, drizzle with more olive oil, cut into wedges, and serve.