Combine oyster sauce, soy sauce, sesame oil, chicken broth, sugar, and cornstarch in a small bowl, whisk until smooth and set aside.
Cook chow mein noodles according to package instructions, drain, rinse with cold water, and set aside.
Heat 1 tablespoon of oil in a large wok or skillet over medium heat.
Add chicken pieces and sauté until cooked through and golden; remove chicken and set aside.
Add remaining oil to the pan, then add carrot, cabbage, and garlic; stir-fry for 2 to 3 minutes until vegetables soften.
Return chicken and noodles to the pan, pour sauce over, mix well, and cook for 2 additional minutes to combine flavors.
Sprinkle sliced green onions on top and serve the chow mein hot directly from the pan.