Pulse cabbage, grated carrot, minced garlic, and minced ginger in a food processor until the cabbage is finely chopped and the mixture starts to come together.
Add crumbled pressed tofu and soy sauce to the processor and pulse a few more times until the filling is cohesive with no large cabbage pieces.
Set up an assembly 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 center.
Dip a fingertip in water and run it around the wrapper edge. Fold the wrapper: bring the ends or corners together, pinch lightly, then pleat one side and press to seal firmly.
Heat a skillet over medium-high and add enough neutral oil to coat the bottom. Arrange potstickers flat-side down without touching; cook in batches if needed until bottoms are golden brown.
Carefully add 1/3 cup water to the pan and immediately cover. Reduce heat to medium-low and steam for about 6 minutes.
Uncover. If excess water remains and potstickers are sticking, raise the heat to evaporate remaining water and crisp the bottoms as needed. Remove cooked potstickers and repeat with remaining batches.
Serve warm with soy sauce, chili crisp, or your preferred dipping sauce.