Mix tamari with one tablespoon olive oil and pour over chicken breast, in a glass pan, turning chicken to coat. Let marinate for at least 30 minutes (up to 4 hours), while grill is preheating.
Preheat grill to medium-high heat, until it reaches about 400 degrees internally.
Brush cabbages with olive oil and salt, then grill on all sides, about 3 minutes each side. Set aside on side of grill where no flame has been ignited (this is indirect heat cooking.)
Add chicken to hot side of grill, and let sear, cooking on one side for 4 minutes. Then flip chicken breast, and cook for another 8-10 minutes, depending on thickness of chicken, until internal temperature reaches 160 degrees F.
Remove cabbage and chicken from grill, and let rest for 10 minutes to cool a bit. Cut out core from each cabbage wedge, and thinly slice cabbage leaves.
Using a very sharp knife, thinly slice chicken. Combine all salad ingredients in a large bowl.
In a jar, whisk together all dressing ingredients, until smooth and combined.
Pour over bowl with salad ingredients, and toss to combine. Plate, and finish with juice of a half lime, extra peanuts, red pepper flakes, and sesame seeds.