Chicken Shawarma: mix all the ingredients except the oil in a bowl. Place the chicken in the bowl and coat it well, cover it, and place it in the refrigerator for at least 4 hours up to 24 hours.
Preheat your oven to 350°F.
In a large baking dish, spread 1 tablespoon of oil, add the chicken and mix well. Arrange the chicken in one single layer, if possible. Bake uncovered for 1 hour until cooked (no pink in the center).
Add the remaining oil in a large skillet over medium-high heat. Add the chicken slices. Sauté until they are golden brown and crispy. If necessary, add 1 tablespoon of oil if the chicken is too dry.
Garlic Dill Sauce: place all the ingredients in a medium bowl. Whisk them all together until creamy and smooth.
To assemble: spread the garlic dill sauce in the middle of each lavash. Add arugula, romaine lettuce, tomatoes, cucumber, and chicken. Roll up the lavash. Toast the lavash roll in a sandwich press or dry griddle until golden brown and crispy. Serve as a roll or cut it into smaller rounds. Use the remaining garlic dill sauce as a dip.
Notes
If you plan to freeze this dish, follow step 1-3; then let the chicken cool. Add the chicken in a ziplock or air-tight container and freeze it for up to 1 week.