Mix beef with 1 tablespoon soy sauce and 1 tablespoon Shaoxing wine (or dry sherry), toss to coat evenly. Leave to marinate at room temperature for 30 minutes or in the refrigerator for up to 1 hour.
Let the beef marinate. Mix the remaining 3 tablespoons of soy sauce with the cornstarch, stirring to make sure the cornstarch is well dissolved. You can do this with a fork to make certain it's well dissolved. Add the last 3 tablespoons of Shaoxing wine (or dry sherry), oyster sauce, chicken broth, sugar, and sesame oil. Stir all together and set the mixture aside.
Combine white part of the green onions, garlic and ginger in the second bowl and set aside to be used later.
When ready to cook, heat 1 tablespoon oil in large sauté pan or wok. Add to high heat until it starts smoking. Add the broccoli and continue stir-fry about 30 seconds then add the water. Cover the pan with a lid or push foil down against the pan sides to seal in, and reduce heat to medium. Steam the broccoli until it reaches just tender but crispy, about 2 minutes. Transfer the broccoli to a paper towel lined plate.
Pat excess moisture from pan. Turn heat up and add other tablespoon of oil. When oil gets hot and smokes,. Add half of the beef to the pan in one layer. Cook without stirring for about 1 1/2 minutes, or until the beef is nicely browned. Stir and cook for an additional 30 seconds, or just until the beef is lightly cooked but still has some pink left. Transfer it to a plate.
Add another tablespoon of oil to the pan and heat until smoking again. Add the remaining beef to the pan, and let it be undisturbed for 1 ½ minutes. Then add in the mixture of scallion whites, garlic and ginger, constantly stirring it with the beef for approximately 30 seconds.
Return the braised beef and steamed broccoli to the pan with the prepared sauce and green parts of the scallions. Bring to a boil, cooking and tossing constantly until just slightly thickened, about 45 seconds. Transfer everything to a serving platter and serve with rice.