Soak mushrooms in 1 cup hot water 15 minutes; drain, reserving 1/4 cup liquid. Chop mushrooms; set aside. Toss chicken with 1 tablespoon cornstarch and 2 tablespoons soy sauce; set aside.
Mash together remaining 2 tablespoons soy sauce, reserved liquid, black beans, and next 3 ingredients; set aside.
Heat 2 tablespoons oil in a skillet or wok over medium-high heat 2 minutes. Add chicken, and stir-fry 5 minutes. Remove from skillet, and set aside.
Heat remaining 2 tablespoons oil in skillet over medium-high heat 2 minutes. Add onion, and stir-fry 1 minute and 30 seconds; add garlic, and stir-fry 30 seconds. Add mushrooms, and stir-fry 1 minute. Add chicken and cabbage, and stir-fry 30 seconds.
Stir together remaining 1 tablespoon cornstarch and 2 tablespoons water until smooth. Add cornstarch mixture and bean mixture to skillet; bring to a boil. Boil 1 minute.