To marinate the chicken, in a bowl, combine the chicken breast with salt and mix until slightly tacky. Add white pepper, soy sauce, dark soy sauce, Shaoxing wine, egg white, cornstarch, and 1 tablespoon cooking oil and mix until evenly coated.
To prepare the sauce, in a bowl, combine soy sauce, dark soy sauce, oyster sauce, and chicken stock and stir until mixed.
To cook the noodles, in a pot over high heat, bring water to a boil. Add the lo mein noodles and cook until just al dente, about 3 minutes. Drain and rinse briefly under cold water.
To stir-fry the chicken, in a wok over high heat, add 2 tablespoons cooking oil. Add the chicken and stir-fry until just cooked through, about 3 minutes. Add the garlic and stir briefly until fragrant. Remove from the wok.
To stir-fry the vegetables, in the same wok over high heat, add 1 tablespoon cooking oil. Add onion, carrot, and cabbage and stir-fry for about 1 minute until slightly softened.
To finish the lo mein, add 1 tablespoon cooking oil to the wok and add the noodles. Stir-fry for about 30 seconds. Return the chicken and vegetables to the wok and pour in the sauce. Toss until the noodles are evenly coated and heated through. Add bean sprouts and scallions and stir to combine.