For the crust: Whisk the chickpea flour and 1 1/2 teaspoons salt together in a medium bowl. Whisk in 1 1/2 cups water and 2 tablespoons olive oil until smooth. Cover and let stand at room temperature for 2 hours.
Preheat the oven to 425 degrees F with a pizza stone or baking sheet on the bottom rack. Line 2 baking sheets with parchment paper. Heat a large nonstick skillet over medium-high heat and add 2 tablespoons olive oil. Whisk the baking powder into the chickpea mixture and pour half of the batter into the hot skillet, swirling to evenly coat the bottom. Cook until just set, about 2 minutes. Transfer the skillet to the hot pizza stone and bake until the edges and top are browned and crisp, about 5 minutes. Flip the crust in the skillet and bake until the top is browned and crisp, about 3 minutes more. Transfer the crust to a parchment-lined baking sheet. Make another crust with the remaining 2 tablespoons olive oil and the remaining batter.
For the toppings: Brush the crusts lightly with olive oil. Arrange the mozzarella on top and sprinkle with Parmesan. Divide the tomatoes between the two crusts and sprinkle with salt and pepper. Roughly tear the basil leaves and scatter over the top. Bake on two sheet pans, switching the position halfway through, until the cheese is browned and bubbly, 10 to 12 minutes. Drizzle with some olive oil and serve.
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