Whisk together yogurt, seasonings, and oil in a medium bowl. Add chicken and stir to coat. Let marinate 20 minutes.
Preheat broiler. Spread chicken on a rimmed baking sheet and broil until cooked through, 8 minutes. Meanwhile, continue with recipe.
Dab naan with a little butter on each side, spread on a rimmed baking sheet, and place in oven on rack below chicken; cook until toasted and crisped, about 8 minutes.
Cut naan in half crosswise and make sandwiches with naan, chicken, tomatoes, onion, feta, mint, and cilantro. Cut each in half if large.
Note: Nutritional analysis is per sandwich.