Whisk all marinade ingredients in a bowl until the sugar dissolves.
Cut chicken into 2-inch pieces, removing excess fat. Add chicken to the marinade, ensuring it's coated well. Marinate in the fridge for at least 2 hours or overnight.
Heat a large skillet over medium-high heat. Add oil and coat the surface. Add marinated chicken and cabbage.
Cook chicken and cabbage for 2 minutes, then add 2 tablespoons of marinade. Toss for another minute, then add another 2 tablespoons of marinade. Cook for 2-3 minutes until chicken is cooked through. Add more marinade if needed.
Transfer cooked chicken and cabbage to a plate, and top with green onions. Serve hot.
Notes
You can add more vegetables like bell peppers or carrots for extra color and flavor.