To prepare dumplings, drop ginger and garlic through food chute with food processor on; process until minced. Add the asparagus, 1/4 cup green onion pieces, and water chestnuts; pulse 4 times or until chopped. Add mushrooms, 2 teaspoons sesame oil, 1 1/2 teaspoons soy sauce, and 1/4 teaspoon salt; pulse 4 times or until mixture is finely chopped.
Working with 1 wonton wrapper at a time (cover remaining wrappers with a damp towel to keep from drying), spoon about 2 teaspoons mushroom mixture in the center of wrapper. Moisten edges of wrapper with water. Bring 2 opposite corners together. Press edges together with fingertips to seal, forming a triangle. Place on a baking sheet coated with cooking spray while assembling; lightly cover with a towel or plastic wrap. Repeat procedure with remaining wonton wrappers and mushroom mixture.
Add water to a Dutch oven to a depth of 1 inch; bring to a boil. Coat a metal vegetable steamer with cooking spray. Arrange 9 dumplings in steamer, slightly overlapping. Steam dumplings, covered, 8 minutes or until tender. Remove dumplings from steamer; cover and keep warm. Repeat procedure with remaining dumplings.
To prepare sauce, combine 1/3 cup soy sauce and remaining ingredients. Drizzle over dumplings.