Rinse mushrooms well and place in a 1 1/2- to 2-quart pan. Cover with hot water and let stand until soft, 15 to 20 minutes. Lift mushrooms out and trim off and discard tough stems. Slowly pour 2 cups of the soaking water into a glass measure and reserve; discard remaining water with sediment.
Rinse pan and return mushrooms and reserved soaking water to it; add soy sauce, sherry, and sugar. Lightly crush ginger and garlic with the flat side of knife and add to pan.
Cover pan and bring to a boil over high heat; reduce heat and simmer until mushrooms are tender to bite, 15 to 20 minutes. Uncover and boil over high heat until most of the liquid has evaporated, about 15 minutes. If desired, remove ginger and garlic. Stir in sesame oil.
Pour mushrooms into a serving dish and sprinkle with green onions. Serve hot or cool.