Finely dice onion, and shred cabbage.
Heat oil in a large skillet over medium heat.
Add onions, and cook, stirring occasionally, until golden brown.
Add the cabbage, carrot, garlic, and ginger. Cook over medium heat, stirring frequently, until cabbage has cooked down and the filling is soft.
Add cubed tofu, and ¼ cup of soy sauce, and cook, for another 2 min.
Turn off heat and set filling aside to cool slightly.
Line a large baking sheet with parchment paper.
Set up an assembly station with a stack of wonton wrappers, a small bowl of water, and your filling. Place a wonton wrapper in one hand, and fill with about 1 tablespoon of filling (you can try making fuller gyoza once you've got the hang of it).
Dip a finger in the bowl of water, and run your wet finger around the perimeter of the wonton wrapper. Fold in half, and pinch around the edges to seal shut. Place on the lined baking sheet. Repeat.
To cook the gyoza from fresh, heat some vegetable oil in a large skillet over medium heat.
Place gyoza in a single layer in the skillet, leaving sufficient room to flip.
Cook until golden on one side, flip, and cook until golden on the other.
Serve immediately, or at room temperature.
To make dipping sauce, whisk ingredients together in a bowl.
Serve.