In a mixing bowl, combine the pork mince, cabbage, carrot, onion, ginger, garlic, soy sauce, hoisin sauce, and sesame oil.
Place wrappers on the work surface. Place a rounded teaspoon of the mince mixture on half of the wrapper, then brush the edges with cold water. Fold over to enclose the filling. To seal the pleats, pinch the edges together.
In a large nonstick frying pan over medium-high heat, heat half of the vegetable oil and place half of the dumplings in the pan. Cook for one minute on each side, or until lightly browned.
Mix with 1/2 cup cold water. Put the cover on the pan and cook for 4 to 5 minutes, or until the gyoza is just tender. Remove from the heat.
Allow standing for one minute in the pan. Transfer to a platter with caution. Cover with foil to keep warm.
Repeat with the rest of the oil and gyoza.
In a small bowl, combine the ingredients for the dipping sauce. Serve gyoza with dipping sauce and sliced green onions.