Combine the teriyaki sauce, olive oil, ginger and garlic in a medium bowl add the chick cutlets and toss to coat. Marinate for ten minutes or several hours if you have time.
Heat a grill pan or prepare a grill. Grill the cutlets until just cooked through, a few minutes on each side. Transfer to a plate and cool. Slice into 1/4-inch wide strips. Cover, and chill the chicken with any accumulated juices until serving time.
Whisk all the ingredients together until well blended. Even better, put all the ingredients in a bullet-type blender and whiz away.
Combine all the salad ingredients in a large bowl, reserving 1/4 cup green onions for garnishing. Add the crispy wontons and toss to distribute. Add the dressing (start with about half), to taste. Transfer the salad to serving plates. Top each salad with chicken slices, drizzle with a little more dressing and garnish with the remaining green onion slices.
A grill pan is a great tool. If you don't have one, get one. It doesn't have to be top-of-the-line to get the job done.