Prep the sauce: In a small bowl, whisk soy sauce, oyster sauce, rice vinegar, honey, broth, cornstarch, and your heat of choice. Set aside.
Slice everything thin: Cut chicken into thin strips.
Slice cabbage and carrot into thin shreds. Separate green onion whites and greens. Mince garlic and ginger.
Heat the pan: Set a large skillet or wok over medium-high heat.
Add 1 tablespoon oil and swirl to coat.
Sear the chicken: Add chicken in a single layer. Season lightly with salt and pepper. Cook 3–4 minutes, stirring once or twice, until just cooked through and lightly browned.
Transfer to a plate.
Sauté aromatics: Add the remaining 1 tablespoon oil. Add green onion whites, garlic, and ginger. Stir 30 seconds until fragrant.
Cook the vegetables: Add cabbage and carrot.
Stir-fry 3–5 minutes until cabbage is crisp-tender. If the pan looks dry, splash in a tablespoon of water to create a little steam.
Combine and sauce: Return chicken to the pan. Give the sauce a quick stir (the cornstarch settles) and pour it in.
Toss for 1–2 minutes until the sauce thickens and coats everything.
Finish: Turn off heat. Stir in sesame oil and green onion tops. Taste and adjust with a splash of soy for salt, vinegar for brightness, or honey for balance.
Serve: Spoon over rice or noodles.
Sprinkle with sesame seeds and add a squeeze of lime if you like.