How to Make It
Finely chop half of ginger. Cut remaining piece in half; set aside. Heat oil in a large skillet over medium-high. Add mushrooms and cook, stirring occasionally, until browned, about 8 minutes. Add garlic, scallions, mirin, 1 tablespoon soy sauce, and chopped ginger; cook until liquid evaporates, about 1 minute. Let cool.
Combine broth, 2 cups water, reserved ginger, and remaining 1 tablespoon soy sauce in a large saucepan and bring to a simmer.
Place 1 teaspoon cooled mushroom mixture in center of a wonton wrapper. Brush edges with a bit of egg. Fold 1 corner of wonton over filling to meet opposite corner. Press to seal edges. Brush remaining 2 sealed corners of wonton with egg and fold under filling to meet; press to seal. Repeat with remaining wrappers and mushroom mixture, keeping formed wontons covered with a damp cloth.
Bring broth mixture to a boil over high. Remove and discard ginger. Add bok choy and cook until bright green and almost tender, about 3 minutes. Add wontons and simmer until cooked through, about 2 minutes. Top with scallions.