In a large bowl, whisk together 2 tablespoons of olive oil, fresh lemon juice, minced garlic, smoked paprika, cumin, coriander, turmeric, cinnamon, salt, and black pepper.
Add the chicken to the bowl and toss to coat completely. Cover and refrigerate for at least 2 hours, or overnight for best results.
In a small bowl, combine the yogurt, mayonnaise, grated garlic, 1 tablespoon of lemon juice, and a pinch of salt. Whisk until smooth and refrigerate until ready to use.
Preheat a grill or grill pan over medium-high heat. Brush the grates with the remaining 1 tablespoon of olive oil.
Grill the marinated chicken for 6 to 8 minutes per side, or until fully cooked through and slightly charred on the edges. Remove from heat and let rest for 5 minutes.
Slice the grilled chicken into thin strips.
Warm the flour tortillas or pita breads on the grill for about 30 seconds.
Assemble the wraps by spreading a generous spoonful of the creamy garlic sauce down the center of each tortilla.
Top with the sliced chicken, shredded lettuce, sliced red onion, and diced tomato.
Fold the sides of the tortilla inward and roll tightly to enclose the filling. Serve immediately.