In a large bowl, whisk together 2 tablespoons light soy sauce, 1 tablespoon rice wine, sesame oil, and cornstarch.
Add the sliced beef to the marinade, coating all sides, and let rest for 15 minutes.
In a small bowl, whisk together dark soy sauce, 2 tablespoons light soy sauce, sugar, and 1 tablespoon wine. Mix well until the sugar is dissolved.
Heat a wok and add 2 tablespoons of oil. When the oil is hot, add the beef. Stir fry until it is nearly cooked. Remove beef from the wok and drain on paper towels.
Clean out the wok, reserving saving 2 tablespoons cooking juices (or more if desired) to add to the sauce later.
Add 2 to 3 tablespoons oil to the wok. When the oil is hot, add the garlic and ginger and stir fry briefly.
Add the 1/2 cup chopped onions and stir fry until they are tender, then add the bok choy stalks. Stir fry a bit longer, then add the mushrooms and finally the bok choy leaves.
Push the vegetables up to the side and re-stir the sauce and add it to the middle, turning up the heat. Push the vegetables back into the sauce and mix well.
Return the cooked beef to the wok. Mix well and stir in the green onions if using. Serve hot with rice.