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Combine garlic, sumac, red pepper flakes, coriander, cumin, and 1 tsp. salt in a small bowl. Arrange tomato slices in a single layer on a rimmed baking sheet and sprinkle garlic mixture evenly over top. Let sit at room temperature at least 30 minutes and up to 1 hour.
Toss chickpeas and vinegar in a medium bowl; season with salt and pepper. Mash chickpeas with a fork until about half are smashed. Add shallot, parsley, and oil and toss to combine; season with salt and pepper. Set aside.
Combine cucumber, yogurt, mint, and parsley in a medium bowl; season with hot sauce, salt, and pepper. Let sit at room temperature at least 15 minutes.
Do Ahead: Yogurt sauce can be made 1 hour ahead. Cover and chill.
Whisk baking powder, sugar, 2 cups flour, and 2 tsp. salt in a large bowl. Add yogurt and stir to combine (the yogurt’s acidity ensures a tender texture). Transfer dough to a lightly floured work surface and knead until mostly smooth, about 1 minute. Divide into 4 pieces, cover with plastic wrap, and let rest 15 minutes (this will make dough easier to roll).
Working one at a time, roll out each piece of dough into a round about ⅛" thick. (Don’t stress: They don’t need to be perfect.) Heat 1 Tbsp. oil in a large skillet, preferably cast iron, over medium. Working one at a time, cook rounds of dough until underside is golden brown and flatbread is beginning to puff, about 2 minutes. Flip and cook until other side is golden brown, about 1 minute longer; transfer to a plate. Repeat with remaining oil and rounds of dough. Let cool slightly.
Top each flatbread with yogurt sauce, chickpea mixture, and tomatoes. Drizzle with more oil and season well with salt and pepper.