Cover stir-fry vegetables in a microwave-safe bowl with 2 tablespoons water. Cook on HIGH for 6 to 8 minutes. Drain and set aside.
Heat sesame oil in a large skillet. Add mushrooms and saute for 2 minutes. Add stir-fry vegetables, garlic, and cilantro. Saute for 2 to 3 minutes. Set aside to cool.
Squeeze out excess moisture from cooled vegetable mixture and place in food processor. Add two egg whites and cornstarch. Pulse until mixture is finely chopped and holds together.
Working in batches, lay out 6 to 8 gyoza wrappers. Lightly brush with the remaining 1 egg white and spoon 1 tablespoon of vegetable mixture in the center of each. Fold in half and press together to seal edges. Stand dumplings seam-side up by tapping on countertop. Repeat with remaining wrappers and filling. Line bamboo steamer with a piece of parchment paper that has been cut to fit. Spray parchment lightly with cooking spray. Put 10 to 12 dumplings in steamer at a time. Place steamer over medium pot of boiling water. Steam for 10 to 12 minutes.
For the Ginger-Chive Ponzu:
In a small bowl, stir together all ponzu ingredients.
Serve steamed dumplings hot with ponzu dipping sauce.