In a small bowl, mix together soy sauce, rice vinegar, sugar, powdered ginger, minced garlic and black pepper. Set the sauce aside.
In a skillet, heat vegetable oil over medium heat. Add the chopped red onion and cook until softened.
Add the chopped mushrooms to the skillet and cook until they release their moisture and start to brown. Stir in the shredded cabbage and grated carrot. Cook for a few minutes until the vegetables are wilted.
Pour the sauce over the cooked vegetable mixture. Stir and cook for another 1-2 minutes until the flavors are well combined. Remove the skillet from the heat and let the filling cool slightly.
To assemble the gyoza, place a gyoza wrapper in the palm of your hand. Place about 1 tablespoon of the cooled vegetable filling in the center of the wrapper.
Moisten the edges of the wrapper with water using your finger. Fold the wrapper in half over the filling to create a half-moon shape. Pleat the edges of the wrapper to seal the gyoza. Repeat with the remaining wrappers and filling.