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Special equipment: a pizza stone or baking sheet; a pizza peel or wooden cutting board
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For the dough: Whisk together the flour, yeast and 2 teaspoons salt in a large bowl.
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Combine 2 1/4 cups lukewarm water and the oil in a second large bowl.
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Add the water mixture to the flour mixture, and combine them with your hands until the dough comes together into a rough and shaggy ball.
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Knead the dough on a flour-dusted work surface until smooth and elastic, about 5 minutes.
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Clean 1 of the large bowls, and lightly coat it with oil.
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Place the kneaded dough in the bowl, brush it with oil, cover and set aside in a warm place until doubled in size, about 1 hour.
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Position an oven rack in the bottom position of the oven, and put a pizza stone or baking sheet on it.
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Preheat the oven to 450 degrees F.
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Transfer the dough to the work surface, and punch it down.
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Cut it into 12 even pieces.
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Shape each into a ball.
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Evenly space the dough balls on the work surface, cover and let rest for 20 minutes.
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(To make ahead, after shaping the dough into balls, wrap each piece tightly in plastic wrap and refrigerate overnight, then let sit at room temperature for 30 minutes before rolling.)
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Working with a few balls at a time, roll each out into a thin round (about 1/8 inch thick), flouring the work surface and rolling pin as needed.
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Sprinkle a pizza peel or wooden cutting board with semolina or cornmeal, and put 1 or 2 pieces of rolled dough on top.
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Shake the peel back and forth to make sure the dough rounds are not sticking, then slide them onto the hot pizza stone and bake until they puff up and are just golden on the bottom but still soft and pliable, about 3 minutes.
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Remove the dough with tongs, press any bubbles down and let cool.
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Repeat with the remaining balls of dough.
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(The parbaked flatbreads can be topped and baked at this point or cooled, stacked, wrapped and refrigerated for up to 2 days or frozen for up to 2 months.)
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For the topping: While the flatbreads bake, heat 2 tablespoons of the oil in a large skillet over medium heat.
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Add the onions and bell peppers, and cook, stirring, until softened, about 5 minutes.
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Add the tomato paste, and stir for 1 minute (add a splash of water if the mixture sticks).
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Add the remaining 4 tablespoons oil, the chicken, cumin, 1 1/2 teaspoons salt, cinnamon, red pepper flakes and allspice, and continue cooking, breaking up the meat, until cooked through, about 5 minutes more.
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Stir in the tomatoes, and cook for 1 minute.
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Remove the skillet from the heat, and stir in the yogurt, lemon juice and parsley .
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Let cool completely.
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(The topping can be made a day ahead and refrigerated.)
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For the sauce: Whisk together the yogurt, 3 tablespoons water and a large pinch of salt in a small bowl.
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To assemble, bake and serve: Brush each parbaked flatbread generously with oil, and top with about 1/4 cup of the topping.
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Bake them, 1 or 2 at a time, on the hot pizza stone until the bottoms are golden and crispy, about 5 minutes, rotating the flatbreads 180 degrees halfway through so they cook evenly.
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Drizzle the flatbreads with yogurt sauce, sprinkle with salt and parsley, cut into slices and serve.
