Heat 1 tablespoon vegetable oil in a large skillet or wok over high heat until shimmering.
Add chicken pieces in a single layer and cook without stirring for 2-3 minutes until golden brown on one side, then stir-fry until cooked through, about 5 minutes total. Remove chicken to a plate.
Add remaining 1 tablespoon vegetable oil to the pan, then add cabbage, garlic, and ginger. Stir-fry for 5-7 minutes until cabbage is tender but still slightly crisp.
Return chicken to the pan and add soy sauce, sesame oil, and red pepper flakes. Toss everything together for 1-2 minutes until well combined and heated through.
Remove from heat, garnish with sliced green onions, and serve immediately over rice or noodles if desired.