Select mushrooms with unopened caps. Cut off stems of mushrooms, so that only caps remain. Retain mushroom stems for soup. Mince the slice of ham, also mince parsley. Assemble other ingredients.
Place oil in wok, heat over low medium heat about 2 minutes. Add mushrooms to wok and stir fry 1 minute. Add 2 tablespoons water, cover wok, and gently stir fry mushrooms for about 6 minutes. Lift cover 2 or 3 times to stir mushrooms during this cooking period. When mushrooms are cooked, remove them to a platter. Arrange mushrooms with caps upright, pour juices from wok over them.
Place 1/4 cup of milk in wok, add thoroughly stirred corn starch solution. Cook over low medium heat until white sauce is thickened. Pour white sauce over mushroom caps, garnish with the minced ham and parsley.