Combine first 3 ingredients in a small bowl. Working with 1 won ton wrapper at a time (cover remaining wrappers with a damp towel to keep them from drying), spoon about 1 tablespoon chicken mixture into center of each wrapper. Moisten edges of wrapper with water; bring opposite corners together. Press edges together to seal, forming a triangle. Cover lightly and set aside.
Combine chicken broth and soy sauce in a large saucepan; bring to a simmer over medium heat. Add 4 dumplings; stir gently. Cook 4 minutes or until dumplings float to top. Place dumplings in a soup bowl. Repeat procedure with remaining dumplings. Add mushrooms and green onions to broth; cook 1 minute. Ladle 1 cup broth mixture over each bowl of dumplings. Serve immediately.