In a small bowl, soak mushrooms in 2 cups boiling water until softened, using a plate to keep mushrooms submerged, about 15 minutes. Remove mushrooms, reserving mushroom liquid, and thinly slice. In another small bowl, whisk together cornstarch and 4 tablespoons water.
In a medium saucepot, add mushrooms and mushroom liquid to chicken broth. Pour mushroom liquid carefully to leave behind grit (or strain it). Bring chicken broth, mushrooms, mushroom liquid, soy sauce, and pepper to a simmer covered. Add bamboo shoots, tofu, and scallions, reserving 4 tablespoons sliced green tops; simmer 5 minutes. Add cornstarch mixture and bring to a boil; cook 2 minutes. Stir in vinegar. Remove from heat and add egg while stirring soup. Serve topped with remaining scallion.