Soak black mushrooms in hot water to cover 15 minutes; drain. Chop and set aside. Chop bamboo shoots, and set aside.
Cut chicken into 1/4-inch-thick strips; toss with 1/4 cup soy sauce and 1 tablespoon cornstarch.
Heat 1/4 cup peanut oil in a Dutch oven or wok at medium-high heat 2 minutes. Add chicken; stir-fry 5 minutes or until done. Remove from Dutch oven, and chop.
Stir together 2 tablespoons soy sauce, 1 tablespoon sesame oil, salt, and pepper. Add to skillet with mushrooms, bamboo shoots, and bean sprouts, and stir-fry 2 minutes.
Stir together cornstarch and 2 tablespoons water until smooth. Stir chicken and cornstarch mixture into mushroom mixture, and stir-fry 2 minutes or until thickened. Remove from heat.
Spoon about 1/3 cup chicken mixture onto center of each wrapper. Fold 1 corner over filling. Fold both adjacent corners over filling. Stir together egg and 2 tablespoons water; brush remaining corner with egg mixture. Roll up, starting at opposite filled side; press edges of brushed corner to seal.
Pour vegetable oil to a depth of 4 inches in Dutch oven or wok; heat to 375°. Fry rolls, a few at a time, 1 to 2 minutes on each side or until golden, and drain on paper towels. Serve immediately with sweet-and-sour sauce.