Marinate the Chicken Toss the sliced chicken in a bowl with water, oyster sauce, cornstarch and vegetable oil. Stir well, so that liquid is well absorbed. Let it rest for 10 minutes.
Make the Sauce: Mix the warm chicken stock with the soy sauce, dark soy sauce, oyster sauce, sesame oil, sugar and white pepper in a small bowl or measuring cup. Mix, then aside.
Blanch the Broccoli: In a large pot bring a pot of water to a boil and add the broccoli and cook for 1 minute (2 minutes for softer broccoli). Drain and set aside.
Cook the Chicken: Heat a wok over high heat until smoking. 2 tablespoons vegetable oil and stir-fry the marinated chicken until it’s no longer pink, about 3 minutes. Transfer to a wok or skillet, and set aside; the chicken will cook through later.
Sauté the Aromatics: Without washing the wok, reduce the heat to medium. Add the remaining 1 tablespoon vegetable oil, then the garlic and ginger. Stir fry for about 5 seconds until it begins to smell fragrant then add the Shaoxing wine around the edge of the wok. Stir in the sauce, and immediately stir into the wok to deglaze. Let it simmer briefly.
Thicken the Sauce: Combine the cornstarch slurry again and pour it in the simmering sauce gradually while constantly stirring. After the mixture boiled, let it cook until the it had become glossy and had thickened, 10 to 15 more seconds.
Combine and Finish: Add the chicken (and any juices) back to the wok, along with the blanched broccoli. Toss to coat all the ingredients with the thickened sauce.
Adjust and Serve: Adjust the seasoning and custom of the sauce if necessary—dark soy sauce for a darker colored sauce, more cornstarch slurry for a thicker sauce, or a little bit of chicken stock if the sauce is too thick. And serve at once with steamed rice.