01 - In a large bowl, whisk together olive oil, yogurt, lemon juice, garlic, cumin, coriander, paprika, turmeric, cinnamon, allspice, cayenne, salt, and pepper until thoroughly combined.
02 - Add chicken thighs to the marinade, tossing to coat completely. Cover and refrigerate for at least 1 hour, preferably up to 6 hours for optimal flavor penetration.
03 - In a food processor, combine garlic cloves and salt. Pulse until finely minced. With the motor running, very slowly drizzle in oil, alternating with lemon juice and ice water, until a thick, creamy emulsion forms. Adjust seasoning as needed.
04 - Preheat grill, grill pan, or skillet over medium-high heat. Remove excess marinade from chicken and grill for 5 to 6 minutes per side, or until cooked through and charred in spots. Let rest for 5 minutes, then slice thinly against the grain.
05 - Warm pitas. Spread each with garlic sauce, top with sliced chicken, lettuce, tomato, onion, and parsley. Roll or fold and serve immediately.