In a medium bowl whisk the yogurt, mint, oil, lemon juice, salt, and pepper. Add the tomato and cucumber and stir to combine.
In a large bowl whisk the marinade ingredients. Add the chicken to the bowl and toss to coat evenly. Marinate at room temperature for 30 minutes.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Lightly brush the onion slices on both sides with oil. Remove the chicken from the bowl, letting the herbs cling to the chicken. Discard the marinade. Grill the chicken and onion slices over direct medium heat, with the lid closed, until the meat is firm to the touch and the juices run clear, and the onion is tender, 6 to 8 minutes, turning once. Remove from the grill.
Warm the flat breads over direct medium heat until lightly charred, about 2 minutes, turning once or twice.
Layer some chicken, sauce, and onion inside the flat breads and serve warm or at room temperature.