Marinate the Chicken: In a bowl, combine olive oil, garlic, cumin, coriander, paprika, turmeric, cinnamon, black pepper, salt, and lemon juice. Add chicken and coat well. Cover and refrigerate for 1–2 hours.
Prepare the Pickled Veggies: In a small saucepan, combine vinegar, water, sugar, and salt. Heat until sugar dissolves. Place cucumber, onion, and carrots in a bowl and pour the hot pickling liquid over them. Let sit for at least 15 minutes.
Cook the Chicken: Preheat a grill, grill pan, or skillet over medium-high heat. Cook chicken for 5–7 minutes per side or until fully cooked. Remove and let rest for 5 minutes. Slice into thin strips.
Warm the Flatbreads: Heat flatbreads on a dry skillet or in the oven until soft and pliable.
Assemble the Wraps: Spread Greek yogurt or tahini sauce over the flatbread. Layer sliced chicken, pickled vegetables, fresh herbs, and optional lettuce or tomatoes.
Wrap and Serve: Fold the sides of the flatbread over the filling and roll tightly. Serve immediately while warm.