Rinse chicken and pat dry. Trim any fat and remove the tender from each breast (reserve for another use). Working 2 at a time, arrange chicken between layers of plastic wrap. With a meat mallet or rolling pin, pound meat to an even thickness of 1/4 in.
Prepare a grill for direct high heat (about 500°; you can hold your hand 5 in. above cooking grate only 2 to 3 seconds). If using charcoal, spread half-lit coals in firegrate, top with cooking grate to preheat, and let heat build to high.
In a medium bowl, combine yogurt, cumin, lemon juice, 3/4 tsp. salt, the chile flakes, and turmeric. Set aside 1/2 cup yogurt mixture. Add chicken to bowl, flip to coat on both sides, and let sit at room temperature 10 minutes. Transfer chicken to a plate, wiping off excess marinade; rub lightly with oil on both sides.
Using metal tongs, wipe cooking grate with oiled paper towels. Grill chicken, covered, 3 to 4 minutes. Turn over, cover, and cook 2 to 3 minutes more. Transfer to a clean plate and season with salt.
In a serving bowl, combine cucumber, shallot, mint, reserved 1/2 cup yogurt mixture, and salt and pepper to taste. Serve chicken with cucumber salad.
Note: Nutritional analysis is per serving.