1Place the oils in a frying pan over medium heat. Add the mushrooms and cabbage and sauté for 1-2 minutes. Remove from heat and combine in a bowl with all the remaining ingredients (except the wrappers and frying oil).
2Place a tablespoon of filling on a gyoza wrapper and lightly brush half the edge with water. Fold the dry half over the wet side and pinch them together, making 5 or 6 pleats to seal the pouch. Continue with the remaining filling and wrappers.
3To cook, place 2 tablespoons of oil in a frying pan over medium to high heat. Add the gyoza to the pan, and working in batches, fry on one side for 2-2.5 minutes until golden brown. Then add 3 tablespoons of water to the pan, cover with a lid and steam for 2-3 minutes, or until the water has evaporated. Continue to cook the remaining gyoza this way, adding more oil as required.
4For dipping sauce, combine all ingredients, whisk, and pour into a serving dish.