Prepare the Gyoza skins. Combine all the ingredients for the skins, stirring with a spatula. Once the dough comes together, knead it by hand for a couple of minutes. Wrap with plastic wrap and let rest for 30 minutes.
Meanwhile, prepare the filling. Finely mince the cabbage and sprinkle with 1/8 teaspoon salt, let sit for a few minutes, then remove excess moisture with a paper towel. Finely mince the brown onion as well. Grate the ginger root.
In a large bowl, combine the prepared cabbage, onion, ginger and the remaining filling ingredients. Mix well.
Mix all the ingredients for the dipping sauce in a small bowl and set aside.
Divide the Gyoza dough into 15 pieces, then roll out into 3-inch rounds, dusting with flour or starch to prevent sticking. Wrap about 1 teaspoon of filling in each skin, making 15 pieces.
Bring a large pot of water to a boil. Cook the Gyoza for 4-5 minutes. Strain and serve in a dish with some of the cooking water to avoid sticking. Serve with the dipping sauce.