Juice 1 of the limes into a medium bowl. Whisk in the cumin, turmeric, 1 cup of the yogurt, 1 teaspoon salt and a generous amount of black pepper. Add the chicken tenders and toss until well coated. Cover the bowl and refrigerate for at least 1 hour and up to overnight.
Meanwhile, cut the remaining lime in half and juice one of the halves into a small bowl. Whisk in 1/2 cup yogurt and 1 tablespoon water until smooth; refrigerate until ready to serve. Cut the remaining lime half into wedges and set aside for serving.
Put the flour, remaining 1 cup yogurt and 2 tablespoons water in a large bowl and mix with a fork until a shaggy dough forms. (The dough may appear dry and crumbly at first, but it will come together as you mix it.) Turn the mixture out onto a lightly floured surface and knead until the dough is smooth and slightly elastic, dusting with more flour if necessary, about 8 minutes. If the dough feels too dry, add water 1 tablespoon at a time and continue kneading until soft and pliable. Divide the dough into 4 pieces, roll into balls and pat out into four 8-inch rounds.
Spray a grill pan generously with nonstick spray and heat over medium heat. Rub 2 of the dough rounds on each side with 1 tablespoon of the oil and place on the grill pan. Cook until dry around the edges and dark grill marks form, about 3 minutes. Flip and cook until golden on the other side, about 2 minutes more. Set aside and repeat with the remaining dough rounds and oil. Brush any remaining dough bits to the sides of the grill pan.
Shake off as much marinade as possible from the chicken pieces and transfer to the grill pan. Cook until dark grill marks form and an instant-read thermometer inserted in the thickest part registers 165 degrees F, about 4 minutes per side.
Spread each flatbread with 1 tablespoon of the yogurt sauce. Top each with 2 chicken tenders and sprinkle with one-quarter of the cucumbers, tomatoes, feta, red onion and cilantro. Drizzle with the remaining yogurt sauce and season with salt and pepper. Fold in half like a taco and serve immediately with lime wedges and mixed greens.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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