Put the turkey breast or thighs into a shallow dish or large, heavy-duty zip-lock bag, add the oil, lemon juice, garlic, spices, salt and pepper and combine with your hands (or a spatula), or seal the bag and knead it to combine the marinade ingredients and coat the turkey well. Refrigerate for a few hours, or overnight. (Or freeze for longer storage.)
Cook on a preheated grill, turning as needed and brushing with the excess marinade (stop brushing it at least 5 minutes before it finished cooking), with the lid closed to trap the heat, for 30-40 minutes, or until the internal temperature reaches 170F. Set aside to rest for at least 10 minutes.
To make the flatbread dough, combine the flour, baking powder and salt in a medium bowl, stir in the oil and then the water, stirring until the dough comes together. Turn out onto the countertop and knead for a few minutes, until smooth. Set aside (cover with a tea towel or plastic wrap, if you like) to rest for about 20 minutes. When you’re ready to cook them, cut the dough into 6-8 pieces, roll out as thin as you can on a clean countertop (don’t flour it, or the dough will slide around) and cook on a preheated grill or in a hot cast iron skillet, with a drizzle of oil, turning as the dough puffs and each side gets grill-marked.
To make the sauce, whisk together the yogurt, tahini, lemon juice, garlic and salt. Add a bit of water if it’s too thick. (Can be made up to a few days in advance—it will be better after a day in the fridge.)
Thinly slice the turkey and serve in warm flatbreads, topped with chopped cucumber, tomato, purple onion, pickles and parsley, and drizzled generously with the yogurt-tahini sauce. Serves 6.