Ribbon cucumber with a vegetable peeler; discard core and tough outer skin. Toss ribbons in a bowl with coleslaw mix, onion slices, vinegar, chopped pickled ginger, ginger juice, and sesame oil to coat. Let stand 10 minutes, stirring occasionally, for flavors to combine.
Cut lavash into 4 equal pieces, and line each with one-fourth of sliced chicken; top with one-fourth of salad mixture. Bring sides of lavash together, and spear with wooden picks. Serve.