For the chicken: Toss the chicken thighs with olive oil, lemon zest and juice, coriander, paprika, allspice, cardamom, cinnamon, ginger, nutmeg, crushed red pepper, grated garlic, 1 teaspoon salt and 1/4 teaspoon black pepper. Cover and marinate in the refrigerator for 2 hours. Remove to room temperature 30 minutes before cooking .
For the fillings: Meanwhile, put the turnips and beets in a bowl or storage container. Combine the vinegar, bay leaf, sliced garlic, 2 teaspoons salt and 1 1/4 cups water in a small saucepan. Bring to a boil and simmer 1 minute. Pour over the turnips and beets to cover. Let cool and refrigerate, 1 hour. It should be ready to eat within an hour.
Whisk together the mayonnaise, yogurt, 1 tablespoon lemon juice, 1 finely grated garlic clove and a pinch of salt in a medium bowl. Reserve.
Whisk together the tahini, the remaining 2 tablespoons lemon juice and grated garlic clove, 3/4 teaspoon salt and 1/3 cup water in another medium bowl. Reserve.
Preheat a grill or grill pan over medium-high heat. Lightly oil the grates. Remove the chicken from the marinade (discard the marinade) and grill the chicken until cooked through with dark grill marks and an instant-read thermometer inserted into the center registers 170 degrees F, 6 to 8 minutes per side. Remove to a cutting board and slice very thinly across the grain. Place in a bowl and toss with 1/2 teaspoon salt.
Lightly warm the pitas on the grill, about 10 seconds per side.
Assemble the wraps: Have 8 sheets of aluminum foil or parchment paper ready. On a pita round, smear a heaping tablespoon of garlic mayonnaise along the middle, then top with a heaping 1/3 cup sliced chicken, a couple of pickled turnip batons, a heaping tablespoon tahini sauce, hot sauce (if using), diced tomato, shredded lettuce, and chopped parsley. Roll the pita tightly around the filling. Wrap with a sheet of foil to hold the roll together. Repeat with the remaining pitas.
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