Combine beaten egg, ground pork, cabbage, ginger, garlic, 2 tablespoons soy sauce, 1 tablespoon mirin, and 1/2 teaspoon sesame oil in a medium bowl. After mixing, place in refrigerator for 30 minutes.

When working with the wraps, cover non-stuffed and stuffed ones with cold, wet paper towels to keep them from drying out. Holding the wrapper in one hand, dip your index on your free hand in cold water. Moisten 1/2 of the wrap with that finger making a 180-degree circle around the edge of the wrap that is furthest from you. Now place a rounded teaspoonful of the meat mixture in the middle. Fold over and gently seal one end to the other, pressing the wrap together to close tightly. Repeat until the meat mixture is gone, placing them on a lightly greased cookie sheet and covering them with moistened paper towels.

Add 2 tablespoons of vegetable oil to a large skillet over medium heat. Add a batch of about ten dumplings and cook until browned on one side, approximately 2-3 minutes. Add 2/3 cup of water to the skillet, cover, and cook for 3 minutes. Uncover the pan flip the dumplings, and cook until the water is gone and the other side of the dumpling has browned; approximately 3-5 minutes.

Mix 2/3 cup soy sauce, 1/3 cup mirin, and 1/8 teaspoon sesame oil in a small bowl to serve with the dumplings. Sprinkle several tablespoons of chopped green onion on the sauce.

Sprinkle the remaining green onions on the dumplings right before serving.