Cut the steak with the grain into 2-inch-wide pieces, then slice thinly across the grain. Combine 3 tablespoons water, 2 tablespoons vegetable oil, 2 tablespoons soy sauce, and 2 teaspoons cornstarch in a large bowl. Add the steak and stir to coat. Refrigerate at least 30 minutes.
Combine 6 tablespoons broth, the remaining 2 tablespoons soy sauce and 6 teaspoons cornstarch, the wine, oyster sauce, sesame oil and sugar in a small bowl, stirring to dissolve the cornstarch completely.
Heat 2 tablespoons vegetable oil in a wok or heavy large skillet over high heat. Add the steak with its marinade and stir-fry until no longer pink, about 2 minutes. Transfer to a platter.
Add 2 tablespoons vegetable oil to the wok. Add the ginger and stir until aromatic, about 30 seconds. Add the broccoli and stir-fry 1 minute. Add 1 cup broth, cover, reduce the heat and simmer 2 1/2 minutes. Transfer the broccoli to a bowl.
Add 1 tablespoon vegetable oil to the wok. Add the mushrooms and cook 2 minutes. Return the steak and broccoli to the wok. Stir the sauce, add to the wok and stir until it thickens, about 30 seconds. Serve immediately with steamed rice.