Whisk together first 5 ingredients. Toss with chicken and marinate, covered, in refrigerator at least 1 hour to overnight.
Oil a large cookie sheet with 1 Tbsp olive oil. Preheat oven to 425F. Cut onion into wedges and toss with chicken and marinade. Transfer chicken and onions from marinade to cookie sheet, arrange in a single layer.
Roast 30-40 minutes until browned and crisping. Slice and serve on flatbread with yogurt sauce, raw onions and tomatoes, fresh herbs, or toppings of your choice.