Season the chicken fillet with salt and ground pepper. Heat the vegetable oil in a skillet over medium-high heat. Fry the fillet on each side until cooked through (internal temperature 74 °C / 165 °F). Set aside to cool.
Finely chop the cabbage. Transfer to a bowl, lightly salt, and stir.
Cut the tomato into thin slices and the cucumber into strips.
In a separate bowl, combine the mayonnaise and ketchup. Add ground pepper to taste and stir.
Lay the lavash on a work surface. Spread the sauce in the center.
Layer the cabbage in the center on top of the sauce.
Add the cucumber and tomato slices on top.
Cut the cooled chicken fillet into pieces and place over the vegetables.
Pour any remaining sauce over the filling.
Roll the lavash tightly into an envelope shape.
Heat a dry skillet over medium-high heat. Place the assembled shawarma in the skillet and cook until golden brown on both sides.