In a bowl, toss the cubed chicken with 1 teaspoon cornstarch and let it marinate for 10 minutes.
In a separate small bowl, combine soy sauce, oyster sauce, sugar, white pepper, sesame oil, 2 tablespoons water, and 1 teaspoon cornstarch; stir until smooth and set aside.
Heat the wok or large frying pan over medium-high heat and add the oil.
Add the peeled ginger slices and stir-fry until fragrant, about 30 seconds.
Add the marinated chicken and stir-fry until the exterior turns white and it is mostly cooked through, about 2–3 minutes.
Add the mushrooms and zucchini to the wok and continue stir-frying until vegetables begin to soften, about 1–2 minutes.
Pour the prepared sauce into the wok and stir quickly to coat the chicken and vegetables; cook until the sauce thickens and the chicken is fully cooked, about 1 minute.
Taste and add salt if needed, then transfer to a serving dish and serve immediately.