Stir together all the Shawarma ingredients (except the chicken and olive oil) in a large bowl. Add the chicken, toss to coat. Cover and refrigerate for 1 hour and up to 1 day.
Heat the oven to 200°F, place a baking sheet in the oven.
Stir together the sweet potato ingredients (other than the oil).
Heat about ½ inch of oil in a skillet.
Grab a handful of the sweet potatoes, form into a patty, making sure to press it together well. Place in the pan (two or three patties may be able to fit into the pan at a time).
Cook unit the underside is browned (flipping too soon will result in the patties falling apart) then carefully flip the patties, press the top with the flat side of the spatula. Cook until browned on the other side.
Remove the cooked patties and place them on the baking sheet in the oven until all the patties are done.
Heat the remaining olive oil in a large cast-iron skillet. Add the chicken, cooking on all sides until browned and cooked through.
Plate the patties, top tomatoes, greens and then with chicken.
Stir together all the aioli ingredients.
Drizzle the chicken with desired amount of aioli, sprinkle with parsley.