Slice chicken breasts into thin strips. In a mixing bowl, combine Greek yogurt, lemon or lime juice, olive oil, minced garlic, and spices. Mix well to coat chicken. Marinate for at least 30 minutes or overnight.
Heat a skillet or grill pan over medium-high heat. Cook marinated chicken strips for 5-7 minutes until cooked through and golden. Use a meat thermometer to ensure chicken reaches 165°F.
Prepare the sauce by mixing sour cream or Greek yogurt, mayonnaise, tahini paste, lemon juice, minced garlic, salt, and pepper in a bowl. Whisk until smooth and creamy.
Lay tortillas, pitas, or saj bread flat. Add cooked chicken, drizzle with sauce, and top with pickled turnips or pickles, sliced onion, optional tomatoes, and chopped parsley.
Fold the sides over the filling and roll tightly. Slice in half if desired and serve with extra sauce for dipping.