To prepare oyster sauce, combine chicken broth through sesame oil in a small bowl. Set aside.
To prepare stir-fry, heat vegetable oil in a wok or large Dutch oven over high heat. Add onions, garlic, and ginger; stir-fry 10 seconds. Add mushrooms and 2 tablespoons sake. Partially cover, reduce heat to medium, and cook 4 minutes. Increase heat to high. Add oyster sauce; stir-fry 4 minutes or until mushrooms are tender. Serve over rice.