Slice boneless skinless chicken breasts into thin strips. Season with salt and pepper to taste.
Heat 1 tablespoon vegetable oil in a large skillet or wok over medium-high heat. Add seasoned chicken strips and stir-fry for 5-7 minutes until cooked through. Remove chicken and set aside.
Add remaining oil to the skillet. Stir-fry garlic, ginger, and the white parts of green onions for about 30 seconds until fragrant. Add shredded cabbage, carrot, celery, and bean sprouts (if using). Stir-fry for 3-4 minutes until vegetables are tender but crisp.
Add yakisoba or chow mein noodles to the skillet. Stir to combine with vegetables.
Whisk together oyster sauce, low-sodium soy sauce, sesame oil, rice vinegar, cornstarch, and brown sugar in a small bowl. Pour sauce over noodle mixture and toss until well coated and heated through, about 2-3 minutes.
Add cooked chicken back to skillet along with green parts of chopped green onions. Stir together and heat for 1 minute. Add Asian Garlic Chili Sauce if desired.
Remove from heat and serve hot, garnished with extra green onions if desired.