Preheat the oven to 400°F. Line a rimmed baking sheet with parchment paper.
Whisk the yogurt, parsley, toum, shawarma seasoning, lemon juice, and olive oil together in a medium bowl. Fold in the chicken to fully coat.
Lay a lavash wrap (or tortilla) flat and add 1 cup of the chicken mixture to the bottom half. Top with 3 tomato slices, 4 pickled turnip slices, and about ½ cup lettuce. Fold the sides over the filling, then roll tightly from the bottom up, like a burrito. Repeat with the other three sandwiches, then set the rolled sandwiches on the prepared baking sheet and bake for 6 to 8 minutes, until golden brown and crispy. (Alternatively, press in a panini press on medium heat for about 6 minutes.) Serve warm with tahini sauce for drizzling.