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To make the dough: In a food processor, process the cottage cheese, water, yeast, sugar, and salt until smooth, 20 to 30 seconds. Add the cilantro, flour, and 1 tablespoon (22g) chili crisp, in that order, and process in about 8 to 10 second-long pulses, until the dough comes together and starts to gather around the blade.
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Scrape the dough into a lightly greased bowl, cover, and let it rise until increased in volume by about 1 1/2 times, 30 to 45 minutes.
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Transfer the dough to a lightly floured work surface and knead it gently, sprinkling with additional flour as necessary, until it's smooth, springy, and less sticky, 30 to 60 seconds.
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Divide the dough into 4 pieces (about 150g each) and shape each into a tight ball. Cover with plastic wrap (or an overturned bowl) and set aside until puffy and relaxed, 15 to 30 minutes.
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To cook the flatbreads on the stovetop: Toward the end of the rise time, preheat a large cast iron skillet over medium-high heat. Working with one dough ball at a time, stretch or roll into a 7" to 8" round. Brush the top with vegetable oil, then place, oil-side down, in the skillet. Cook until brown underneath and bubbled on top, 1 to 2 minutes. Flip the dough and cook until brown on the second side, 1 to 2 minutes. Wrap the flatbreads in a kitchen towel to keep warm, stacking as you go. While one flatbread is cooking, shape the next.
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To broil the flatbreads: Alternatively, cook the flatbreads in the oven. Toward the end of the rise time, preheat the broiler (on “high” if your oven has that option) with a rack in the center. Grease a baking sheet with 1 1/2 teaspoons vegetable oil. Place 2 balls of dough on the baking sheet, turn to coat in oil, then stretch into 7" to 8" rounds. Broil the flatbreads until golden brown on top, 1 to 3 minutes. Flip and broil until evenly golden brown on the second side, 1 to 2 minutes. Wrap the flatbreads in a kitchen towel to keep warm. Let the baking sheet cool, then repeat the process with the remaining 2 balls of dough and another 1 1/2 teaspoons vegetable oil.
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For an extra spicy finish, brush the warm flatbreads with chili crisp before serving.
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Storage information: Store the cottage cheese flatbreads, well-wrapped, at room temperature for 1 to 2 days; rewarm (in a toaster or in a foil packet in the oven) before serving.