Saute green onion and chopped ginger in sesame oil until limp, 2-3 minutes. Set aside.
Boil edamame in water to barely cover until crisp-tender, about 5 minutes. Chop very coarsely and set aside.
Combine tofu, miso, salt (if needed) and green onion mixture until smooth (using hands or food processor). Fold in chopped edamame by hand or with a spatula.
Divide into 9 pieces. Form each into a round or oval pattie and coat in cornstarch. Fry in vegetable oil 1" deep (the oil is hot enough when a small bit of the mixture turns brown in 3-4 seconds). Fry in batches without overcrowding, until browned all over. Drain on paper towel and cool to room temperature.