Freeze the tofu overnight. Defrost, thoroughly wash and squeeze as much water out of the tofu as you can. See my post on "vegetarian ground meat" for more detailed instructions.
Wash the quinoa and cook it in 1 cup of water. Add it to the bowl with the tofu when cooked.
Boil the cabbage until it's not crisp anymore, but not until it's fully soft (about 1-2 minutes). Drain it and run under cold water so it's cool enough to handle. Shake out of the excess water but do not squeeze. Mince the cabbage and add it to the bowl with the tofu and quinoa.
Add the scallions, garlic, ginger, sesame oil, soy sauce, sake, potato starch, salt and white pepper to the bowl. Use a Microplane to grate the dried shiitake into a powder and add it to the bowl.
Put on a food-safe glove and mix the filling with your hand, using a kneading action to coax the crumbly mixture to come together. It will never stick together like a meat-based filling, but it should hold its shape when scooped together with a spoon.
If you're right handed, place 1 wrapper in the palm of your left hand, then place a spoonful of filling into the center of the wrapper.
Dip the fingers of your right hand in a bowl of water, and wet the entire rim of the wrapper. This is what seals the pleats into place.
Continue holding the dumpling in your left hand and fold the wrapper in half but don't seal the edges yet.
Pinch the left edge shut and hold it shut with your left thumb and forefinger.
Fold a pleat with your right thumb and forefinger.
Pinch the pleat shut with your left thumb and forefinger. While using the fingers on your right hand to keep the filling from squeezing out.
Continue pleating and pinching until you've reached the right edge. Repeat until you run out of filling or wrappers. If you're not going to fry them right away, put the dumplings on parchment paper and leave some space between each one to keep them from sticking. You can freeze them like this and transfer them to a freezer bag after they're frozen.
Add 2 tablespoons of oil to non-stick frying pan and heat over medium-high heat. Add the gyoza in rows.
Fry until the gyoza are just starting to turn tan on the bottom. With a lid at the ready, add ¼ cup of water to the pan and immediately cover with a lid. The oil is going to spatter quite a bit so be careful and use the lid as a shield to protect yourself.
Let the gyoza steam for 2 minutes. If the water runs out before the 2 minutes are up, crack open the lid and add a little bit more.
After steaming, remove the lid and let the remaining water burn off. Let the gyoza fry in the remaining oil until the bottoms are golden brown and crispy. Plate with the crispy side up so they don't get soggy.
To make the sauce for the gyoza, combine equal parts soy sauce and rice vinegar and add chili oil to taste.