1. 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.
2. 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.