In a bowl, combine the chicken, 1 tablespoon soy sauce, 1 tablespoon cornstarch, 1 teaspoon sesame oil, and ginger. Mix well and set aside for at least 15 minutes.
Cook the chow mein noodles according to package directions. Drain and set aside.
In a small bowl, whisk together the 1/4 cup soy sauce, oyster sauce, brown sugar, 1 tablespoon cornstarch, chicken broth, and 1 teaspoon sesame oil. Set aside.
Heat 1 tablespoon of vegetable oil in a large wok or skillet over medium-high heat. Add the chicken and stir-fry until cooked through and lightly browned, about 5-7 minutes. Remove the chicken from the wok and set aside.
Add the onion and garlic to the wok and stir-fry until fragrant, about 1 minute. Add carrots and celery, continue to stir-fry for 2 minutes. Add the cabbage and cook until tender-crisp, about 2 more minutes.
Add the cooked noodles and chicken back to the wok with the vegetables. Pour the sauce over the mixture and stir-fry until the sauce thickens and coats everything evenly, about 2-3 minutes. Sprinkle with green onions and bean sprouts (if using). Serve immediately.