Heat 1 tablespoon of sesame oil in a large skillet over medium-high heat. Add cabbage, onion, carrot, and garlic. Cook and stir until the cabbage becomes limp and vegetables soften, about 3-4 minutes.

Add ground pork and beaten egg to the vegetable mixture. Cook and stir until the pork is browned and crumbly, about 5-7 minutes. Stir in ginger, green onions, soy sauce, rice vinegar, and white pepper. Remove from heat and allow to cool for 10-15 minutes.

Place a gyoza wrapper in your palm. Dip your finger in water and run it around the edge of the wrapper. Place about 1 tablespoon of filling in the center. Fold the wrapper in half to create a half-moon shape, then pinch the center of the edge. Create 3-4 pleats on each side by folding the wrapper towards the center pinch. Press firmly to seal.

Place assembled gyoza on a baking sheet lined with parchment paper, making sure they don't touch each other to prevent sticking.

Heat 2 tablespoons vegetable oil in a large non-stick skillet over medium-high heat. Place gyoza flat-side down in a circular pattern and cook until bottoms are golden brown, about 3 minutes.

Carefully add ⅓ cup of water to the pan (it will splatter), then immediately cover with a lid. Reduce heat to medium and steam for 3-4 minutes until the water evaporates and the wrappers become translucent.

Remove the lid and continue cooking for another 1-2 minutes to re-crisp the bottoms. Transfer to a serving plate.

While gyoza are cooking, combine all dipping sauce ingredients in a small bowl and mix well.

Arrange gyoza on a serving plate with the crispy side up. Serve immediately with the dipping sauce.