For the ginger dipping sauce: Combine the soy sauce, vinegar, sesame oil, sugar and ginger in a small bowl and set aside.
For the filling: Mix together the pork, egg, sesame oil, ginger, garlic, scallion, soy sauce, vermouth, cornstarch, sugar and 1/2 teaspoon salt in a bowl and refrigerate for 20 minutes.
To make the gyoza: Place a small bowl of cool water by your work surface. Working with one wrapper at a time and keeping the rest tightly wrapped, put a wrapper on your work surface and brush the edges with water using your finger or a small pastry brush. Put a mounded teaspoon of filling in the center of the wrapper, spreading it out into a football shape. Bring the edges of the wrapper together, resting the dumpling base on your work surface. Seal the edges by making 4 to 6 pleats on the side facing you and pressing the edges together. (Only 1 side should be pleated.) Set aside, cover with a damp paper towel and repeat with the remaining wrappers and filling.
To cook the gyoza: Heat 2 tablespoons of the vegetable oil in a large nonstick skillet over medium-high heat until it shimmers. Quickly arrange half the gyoza, bottom-side down in the skillet, and cook until beginning to brown on the bottom, about 1 minute. Add 2/3 cup water to the skillet (it will spatter), cover tightly and let the gyoza cook until the skins are tender, about 6 minutes. Remove the lid and cook until the liquid has cooked off and the gyoza are crisp and brown on the bottom, about 4 minutes. Repeat with the remaining vegetable oil and dumplings. Serve hot with the ginger dipping sauce.
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