Heat 1 tablespoon of vegetable oil in a large skillet or wok over medium-high heat. Add the sliced chicken and season with salt and pepper. Cook for about 5-7 minutes, or until the chicken is cooked through. Remove the chicken from the skillet and set aside.
In the same skillet, add the remaining tablespoon of vegetable oil. Add the garlic and ginger, sautéing for about 30 seconds until fragrant.
Add the broccoli, bell peppers, and carrots to the skillet. Stir-fry for about 3-4 minutes until the vegetables are tender-crisp.
Return the cooked chicken to the skillet. Add the bean sprouts and green onions, stirring to combine.
In a small bowl, mix the soy sauce, oyster sauce, cornstarch, and chicken broth. Pour this sauce over the chicken and vegetables in the skillet. Stir well to coat everything evenly and cook for an additional 2-3 minutes until the sauce thickens.
Add the cooked chow mein noodles to the skillet, tossing everything together until the noodles are heated through and well combined with the sauce.
Serve hot, garnished with additional green onions if desired.