In a large plastic food storage bag, combine chicken cutlets with teriyaki sauce, 2 tablespoons oil and grill seasoning. Close bag and combine to coat evenly. Let stand 15 minutes.
Preheat an outdoor or indoor grill or grill pan to medium-high.
Cut cabbage into quarters and cut away core. Shred the cabbage and set aside. Cut clean, trimmed scallions into 3 inch lengths. Pile scallions lengthwise and julienne them into thin strips. Pull the threads from the ends of snow peas and julienne into thin strips, lengthwise, like the scallions.
Drizzle honey into a small bowl. Add vinegar and combine with a fork.
Place chicken cutlets on grill or into hot pan and cook 3 minutes on each side.
Heat a nonstick skillet over high heat. Add 2 tablespoons oil, 2 turns of the pan. Add pickled ginger and cut carrots and stir-fry 2 minutes. Add cabbage and stir-fry 2 minutes more. Fluff and toss veggies with tongs so they stay dry and crisp while cooking. Add scallions and snow peas and stir-fry another minute. Add honey and vinegar combination, pouring it all around the pan in a slow stream. Cook the liquids down 30 seconds more then turn off heat. Continue to toss the slaw and season it up with salt and pepper to your taste.
Slice the chicken meat thinly on an angle. Mound up 1/4 of the slaw on each plate and place the sliced chicken along side, edging its way up the slaw salad. Serve immediately.