Soak mushrooms in warm water until soft, about 30 minutes. Drain. Rinse in warm water, drain. Remove and discard stems. Cut caps into 1/2 inch pieces.
Trim fat from beef. Cut grain-wise into 2 inch strips. Cut the strips across the grain into 1/8 inch strips.
Toss the strips with: 3 T vegetable oil, 1 T cornstarch, 1 T salt, 1 T sugar, 1 T soy sauce, and 1/4 tsp white pepper in plastic or glass bowl. Cover and refrigerate for at least 20 minutes.
Cut broccoli into 1 inch pieces. Steam broccoli. Cook to al dente. When done, rinse under cold water and drain. Set aside.
Cut green onions into 1/4 inch rings. Mix together in a bowl green onions, 1/2 cup chicken broth, 1/4 cup oyster (or soy) sauce, and 2 tablespoons cornstarch. Set aside.
Heat wok until 1 or 2 drops of water bubble and skitter when sprinkled into wok. (Or if you're using a skillet, put on medium-high heat.) Add 2 tablespoons oil and rotate to coat wok. (Skillet-1 tablespoon and coat bottom of pan).
Add beef and garlic (mix the garlic with the beef first to prevent burning it). Stir-fry until beef is brown-about 3 minutes.
Remove beef from wok (or skillet). Set aside.
Add mushrooms and broccoli with 1 cup water. Cover for 5 minutes.
Stir in beef. Heat to steaming.
Stir in Sauce Mixture. Cook and stir until thickened, about 15 seconds.