Prep: Slice chicken thin against the grain. Shred cabbage. Whisk coconut aminos and vinegar; set aside.
Sear chicken: Heat 2 tbsp oil in a hot wok/skillet. Add chicken in a single layer and sear 45–60 seconds per side until browned. Remove to plate.
Aromatics & cabbage: Add remaining oil, then garlic and ginger for 20–30 seconds. Add cabbage and toss; press into a layer and let char 1–2 minutes. Stir-fry 3–5 minutes until slightly tender.
Combine & sauce: Return chicken to pan. Pour sauce over and toss to coat. If using, scramble eggs on one side and fold into the mix. Cook 1–2 minutes until hot and glossy.
Finish: Turn off heat; stir in green onions, sesame seeds, and remaining sesame oil. Adjust seasoning. Serve hot.