Rinse mushrooms and place in a bowl; cover with 2 cups hot water. Let soak until soft, about 20 minutes. Lift mushrooms out and squeeze their liquid into bowl; reserve liquid. Cut off and discard tough stems from mushrooms; thinly slice caps.
Carefully pour 3/4 cup of the mushroom-soaking liquid into a 1- to 2-quart pan, leaving sediment behind. Add sliced mushrooms, sherry, soy sauce, garlic, and sugar. Bring to a boil over high heat, then cover, reduce heat, and simmer until mushrooms are tender to bite, about 5 minutes.
In a small bowl, mix cornstarch with 2 tablespoons water; add to mushrooms and stir until boiling.
Arrange hot greens on a platter. Spoon mushroom sauce in a band over the top.