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Prepare the Filling: In a large mixing bowl, combine the shredded cabbage, grated carrot, chopped shiitake mushrooms, green onions, grated ginger, minced garlic, soy sauce, and sesame oil. Add the beaten egg (or flax egg) and mix until well combined. Season with salt and pepper to taste.
Assemble the Gyoza: Place a gyoza wrapper on a flat surface. Spoon a small amount of the vegetable filling into the center of the wrapper. Moisten the edges of the wrapper with water. Fold the wrapper in half over the filling and pinch the edges to seal, creating pleats. Repeat with the remaining wrappers and filling.
Cook the Gyoza: Heat a large skillet with a lid over medium heat. Add a thin layer of vegetable oil. Place the gyoza in the skillet, flat side down, and cook until the bottoms are golden brown, about 2-3 minutes.
Steam the Gyoza: Add about 1/4 cup of water to the skillet and cover immediately. Steam the gyoza for about 5 minutes, or until the water has evaporated and the wrappers are tender and translucent.
Serve: Remove the gyoza from the skillet and serve hot with your preferred dipping sauce.
