Combine the cabbage, carrots, garlic, and ginger in a food processor. Pulse until the mixture starts to come together, and the cabbage is in smaller pieces. Crumble in the tofu and add the soy sauce, pulsing a few times to bring everything together. The mixture should look cohesive with no large pieces of cabbage poking out.
Set up an assembling station with a small bowl of water, filling, and wrappers. Working with one wrapper at a time, place about 1 tablespoon of filling in the middle. Dip your finger in the water and run it along the entire edge of the wrapper. Fold both ends up (or corners if you are using square wrappers) and lightly pinch. Working with one side, pleat and pinch together, making sure that once pleated, the potsticker is sealed. Repeat with remaining filling and wrappers.
Heat a skillet over medium-high heat. Add enough oil to coat the bottom of the pan and add potstickers, so they are not touching, and the flat side is down in the pan. I usually have to work in batches. Cook until the bottoms golden.
Using the lid to shield yourself from the splatter, add ⅓ cup of water to the pan. Cover immediately, reduce the heat to medium-low and cook for about 6 minutes. Once done, lift the lid. If the potstickers seem stuck, turn the heat back up and cook off the water if there is any left. (This is how I cook potstickers with a stainless steel pan.) Remove from the pan and repeat the process with however many more potstickers you would like to cook.
Serve with soy sauce, chili crisp, or your favorite sauce.