Preparation Place a rack in top third of oven; preheat to 450°F. Whisk chickpea flour, 2 Tbsp. oil, 1 tsp. salt, and 1 cup water in a medium bowl or large measuring cup until combined. Let sit at least 10 minutes and up to 1 hour to let flour hydrate. Heat 2 Tbsp. oil in a 12" cast-iron skillet over high until shimmering. Pour in exactly half (3/4 cup) of the chickpea batter and tilt to evenly coat skillet. Transfer to oven and bake flatbread until edges are golden brown, 8–10 minutes. Using a large spatula, transfer flatbread to a wire rack. Repeat with remaining batter. Meanwhile, cook tomatoes, garlic, 1/4 cup oil, and remaining 1/2 tsp. salt in a large skillet over medium-high heat, stirring occasionally, until tomatoes begin to burst, 4–5 minutes. Stir in chickpeas, oregano, vinegar, and paprika (if using). Reduce heat to medium and continue to cook, stirring often, until tomatoes have burst more into a fully formed sauce, 3–4 minutes. Remove from heat and stir in butter. Slice flatbreads in half. Serve with tomato-chickpea mixture, feta, greens, and lemon wedges alongside for squeezing and oil for drizzling.