Combine mushrooms and water in a bowl; let sit until softened, about 10 minutes. Drain mushrooms and squeeze dry. (Save the mushroom soaking liquid: You can add it to your next soup or sauce. Just save it in the fridge in a covered container and find a use for it in about a week—or freeze it for up to 3 months.) Using scissors, starting from the outside perimeter and turning the mushroom, cut one long 1/4-inch strand of mushroom. Don’t worry if it breaks, but the goal is to get it as long as possible.
Heat 2 inches of oil in a flat-bottomed wok or frying pan with sides until it reads 350° F on a deep-fry thermometer. Toss mushrooms with 3 tablespoons of cornstarch, making sure to shake off any excess cornstarch. Working in batches, fry mushrooms until crisp, 1 to 2 minutes. Transfer mushrooms to paper towels to drain. Make a slurry by mixing together 1/2 teaspoon cornstarch plus 1 teaspoon water.
Discard all but about 1 tablespoon oil from wok; heat over medium-high. Cook three-quarters of scallions and the ginger until fragrant, about 30 seconds. Add soy sauce, sugar, and 1 tablespoon water (you could also use vegetable or chicken stock here).
Slowly add slurry; cook, stirring constantly, until thickened, 1 to 2 minutes (if it is too thick, add more water). Stir in reserved mushrooms; cook 1 minute. Transfer to a serving platter; garnish with remaining scallions.
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