Cook the noodles according to the package directions until just tender. Drain them well, then rinse briefly if needed to keep them from sticking.
In a small bowl, whisk together the soy sauce, oyster sauce, chicken broth, rice vinegar, cornstarch, brown sugar, and black pepper until smooth.
Heat a large skillet or wok over medium-high heat. Add a little oil, then cook the sliced chicken until lightly browned and cooked through. Transfer it to a plate.
Add a bit more oil to the pan, then stir-fry the garlic and ginger for a short moment. Add the carrots, bell peppers, and cabbage, and cook until they are crisp-tender.
Return the chicken to the pan with the noodles. Pour the sauce over the top and toss everything together until the noodles are coated and the sauce has thickened slightly.
Stir in the green onions, then taste and adjust with a little more soy sauce or pepper if needed. Serve warm right away.