In a large mixing bowl, combine the Greek yogurt, lemon or lime juice, olive oil, minced garlic, cumin, coriander, smoked paprika, salt, all spice or 7 spice, turmeric, cinnamon, garlic powder, and black pepper. Mix well to create a marinade.
Add the chicken breasts to the marinade, ensuring they are well coated. Cover the bowl with plastic wrap and let it marinate in the refrigerator for at least 1 hour, or ideally overnight, for maximum flavor.
While the chicken is marinating, prepare the sauce. In a separate bowl, whisk together the sour cream or Greek yogurt, mayonnaise, tahini paste, lemon juice, minced garlic, and a pinch of salt and pepper. Adjust seasoning to taste and set aside.
Heat a skillet or grill pan over medium-high heat. Once hot, add the marinated chicken breasts, cooking for about 6-7 minutes per side or until the internal temperature reaches 165°F (75°C). Remove the chicken from the skillet and let it rest for a few minutes before slicing.
Thinly slice the cooked chicken. Lay out your tortillas, pitas, or saj bread, and spread a generous amount of the sauce on each. Top with sliced chicken, pickled turnips or pickles, sliced onion, thinly sliced tomato (if using), and chopped parsley.
Fold the sides of the tortilla or pita over the filling, then roll it up from the bottom to enclose the filling. Secure with a toothpick if needed, and enjoy your Chicken Shawarma Wraps!