In a large mixing bowl, whisk together olive oil, lemon juice, minced garlic, cumin, smoked paprika, coriander, turmeric, cinnamon, allspice, black pepper, and salt until well combined.
Add chicken thighs to the marinade and massage thoroughly to coat each piece. If thighs are thicker than 1 inch, pound slightly for even cooking. Cover and refrigerate for at least 2 hours or up to 24 hours for best results.
Preheat oven to 400°F (200°C).
Line a baking sheet with aluminum foil and arrange marinated chicken thighs in a single layer without overlapping.
Bake for 25-30 minutes, flipping halfway through, until internal temperature reaches 165°F in thickest part and edges are golden-brown with slight charring. For extra crispy edges, move rack to top position and broil for last 2-3 minutes, watching carefully.
While chicken bakes, combine mayonnaise, Greek yogurt, minced garlic, lemon juice, olive oil, and salt in a food processor or blender. Blend until smooth, adding water one tablespoon at a time to reach desired consistency. Sauce should be slightly thinner than desired as it will thicken when refrigerated.
Let cooked chicken rest for 5 minutes until warm but not steaming hot, then transfer to cutting board and slice into thin strips about 1/4 inch thick.
Warm pita bread in oven or skillet for 30 seconds per side, watching carefully to prevent drying. Assemble shawarma with sliced chicken, garlic sauce, tomatoes, cucumbers, onions, parsley, and pickles, or serve as a bowl over rice.
Notes
Chicken thighs stay juicier than breasts. If using breasts, pound to even thickness and reduce cook time by 5 minutes.
Marinate at least 2 hours, up to 24 hours for best flavor penetration.
Let garlic sauce rest in refrigerator for 30 minutes before serving for best flavor.
Freeze cooked chicken up to 3 months. Make fresh garlic sauce when ready to serve.
Always use meat thermometer to check for 165°F internal temperature.