Toss garlic, thyme, mint, red pepper flakes, coriander, cumin, salt, and tomatoes in a bowl. Let sit at room temperature for 30 minutes.
To Make The Marinated Chickpeas:
Place the chickpeas and vinegar in a bowl. Season with salt and pepper. Mash approximately half of the chickpeas with a fork. Stir in shallots, parsley, and olive oil. Set aside.
To Make The Herb Yogurt Sauce:
Mix everything together in a small bowl. Check for seasoning. Let sit for 15 minutes at room temperature.
To Make The Flatbreads:
Whisk baking powder, sugar, flour, and salt in a large bowl. Add the yogurt and stir to combine. Transfer the dough to a floured work surface and knead until smooth, about 1 minute. Divide into 4 pieces, cover with a towel, and let rest for 15 minutes.
Working with one piece of dough at a time, roll out circles of approximately 1/8-inch thick. Heat one tablespoon of oil in a large skillet over medium heat. Cook each flatbread for about 2 minutes. Flip, and cook for an additional minute. Transfer to a plate. Repeat until all the flatbreads are cooked.
To Assemble The Dish:
Top the flatbreads with yogurt sauce, chickpeas, and tomatoes. If desired, top with crumbled feta cheese and mint.