Lay eggplant on parchment-lined half sheet tray and sprinkle with salt; place sheet of parchment and half sheet tray over eggplant. Weigh down with about 2 pounds for 1 hour.
Drain water from eggplant; brush both sides of eggplant slices with oil. Grill eggplant and peppers over medium-high heat 6 minutes or until tender, turning once. Cool; cut eggplant and peppers into ¼-inch-thick slices.
For the feta sauce:
Whisk together all ingredients except the onions in a bowl; fold in onions. Makes about 1 cup.
For the chicken shawarma:
Whisk together all ingredients except the chicken in a bowl; add chicken, turning to coat. Cover and refrigerate at least 1 or up to 3 hours.
Remove chicken from marinade; discard marinade. Grill chicken 10 minutes or until internal temperature reaches 165°F, turning once. Cut chicken into ¼-inch-thick slices.
Heat 1 flatbread directly on oven rack at 400°F for 6 minutes or until warmed. Spread flatbread with
¼ cup feta sauce; top with ½ chicken breast, ¼ cup each eggplant, peppers and tomatoes. Cut flatbread into quarters and place on a plate; place ½ cup greens in center.