Cook edamame according to package instructions.
Reserve 1/2 cup cooking water; drain edamame and set 3/4 cup aside.
In a blender, whirl remaining 1 cup edamame with ginger, lemon juice, salt, and about 1/3 to 1/2 cup reserved cooking water.
Transfer edamame paste to a bowl, mix with whole edamame and chives.
Bring a large, deep pot of salted water to a gentle boil and add vegetable oil.
Fill wonton wrappers with 1 tsp. edamame filling and top filling with a cilantro leaf (cover unfilled wrappers with a moist towel). Moisten wrapper edges with water, fold over filling, and press to seal. Boil wontons in batches until they float to surface, 2 minutes. Serve with soy sauce.