In a large bowl, whisk together the yogurt, lemon juice, olive oil, minced garlic, cumin, paprika, turmeric, coriander, allspice, cinnamon, cayenne pepper (if using), salt, and pepper.
Add the chicken thighs to the bowl and coat thoroughly with the marinade. Cover and refrigerate for at least 30 minutes, or up to overnight.
Preheat oven to 400°F (200°C), grill to medium-high heat, or heat a skillet over medium-high heat.
For oven: Arrange chicken on a baking sheet and bake for 25-35 minutes. For grill: Grill chicken for 5-7 minutes per side. For stovetop: Cook chicken in a skillet with olive oil for 6-8 minutes per side, or until cooked through (165°F/74°C).
Let the chicken rest for 5-10 minutes, then slice thinly against the grain.
Warm pita bread and fill with sliced chicken shawarma. Add desired toppings like tomatoes, cucumbers, red onion, and tahini sauce or hummus.