With a clean chopping board, slice the chicken into medium size cubes. They need to be large enough to be on skewers. About 1 inch cubes.
In a large bowl, combine the yogurt, grated garlic, dill, cumin, turmeric, paprika, lemon juice, salt, and pepper. Add the sliced chicken thighs and toss until well coated. Cover and refrigerate for at least 2 hours, preferably overnight.
Preheat a grill or grill pan over medium-high heat. Remove the chicken from the marinade and thread onto skewers. Grill for 5-6 minutes on each side, or until the chicken is cooked through and slightly charred. Remove from the grill and place it on a plate.
In a large mixing bowl, combine the chopped romaine lettuce, grated Parmesan cheese, Caesar salad dressing, lemon juice, salt, and pepper. Toss until the lettuce is evenly coated with the dressing.
To assemble the wraps, warm the pita bread or tortilla wraps according to package instructions. Place a generous amount of Caesar salad on each wrap, then top with the grilled chicken shawarma slices.
Roll up the wraps tightly, folding in the sides as you go. Cut in half diagonally and serve immediately.