Soak the dried fungus and mushrooms in separate bowls of hot water for 20-30 minutes, depending onthe thickness of the mushrooms. The thick, top-quality mushrooms require longer soaking than those with thin caps. Squeeze out and save mushroom soaking water, trim off stems and cut caps into halves or quarters. Rinse fungus and cut into bite-sized pieces, discarding any tough gritty bits. Thickly slice or halve the button mushrooms or oyster mushrooms.
Heat a wok, add oil and swirl to coat surface. Fry the shiitake mushrooms for a few minutes, then add garlic and ginger, stir-frying until golden. Add 2 cups of the mushroom soaking liquid mixed with soy sauce and sugar, cover and simmer for 20 minutes. Add all remaining fungus and fresh mushrooms, stir well, cover and simmer for 10 minutes. Stir in butter.
Add asparagus and toss over high heat. If liquid has reduced, add 1/2 cup hot water or mushroom water, cover and cook for 2 minutes or until asparagus is just tender. If liked, thicken sauce by stirring in 1 teaspoon cornflour mixed with a tablespoon of cold water until it boils and becomes clear. Serve at once.