Begin by marinating the chicken. In a bowl, combine olive oil, cumin, paprika, turmeric, coriander, cinnamon, cayenne pepper, and season with salt and black pepper. Add the chicken thighs and ensure they are well-coated. Let sit for at least 15 minutes or overnight for more flavor.
While the chicken is marinating, prepare the pickles. In a small bowl, mix the cucumber slices with vinegar, sugar, and salt. Set aside to pickle lightly for at least 15 minutes.
Preheat a grill pan or skillet over medium-high heat. Add the marinated chicken and cook for about 5-7 minutes on each side, or until the chicken is fully cooked and has a nice char. Remove from heat and let it rest before slicing into strips.
For the garlic sauce, whisk together Greek yogurt, tahini, minced garlic, lemon juice, and salt in a bowl until smooth and creamy.
Warm the flatbreads on the grill pan for about 30 seconds on each side.
To assemble the wraps, spread a generous amount of garlic sauce on each flatbread. Add a portion of the chicken strips, top with pickled cucumbers, and roll the flatbread tightly around the fillings.
Serve the chicken shawarma wraps immediately, with extra garlic sauce and pickles on the side if desired.