In a large bowl, combine the yogurt, olive oil, minced garlic, ground cumin, ground paprika, ground coriander, ground turmeric, ground cinnamon, ground allspice, black pepper, salt, and lemon juice.
Add the chicken thighs (or beef/lamb slices) to the marinade and mix until the meat is well-coated.
Cover and refrigerate for at least 2 hours, or preferably overnight, to allow the flavors to penetrate the meat.
Preheat a grill or grill pan over medium-high heat. If using a skillet, heat a little oil over medium heat.
Remove the meat from the marinade and cook it on the grill or skillet until fully cooked and slightly charred, about 5-7 minutes per side for chicken, or until the meat is cooked through.
Let the cooked meat rest for a few minutes before slicing it thinly.
In a small bowl, mix together the mayonnaise, plain yogurt, minced garlic, lemon juice, olive oil, and salt.
Warm the pita bread or flatbreads in the oven or on a skillet.
Place the sliced Shawarma meat on the warm bread.
Top with sliced tomatoes, cucumbers, onions, and fresh parsley.
Drizzle with garlic sauce if desired.
Enjoy your meal!