In a large bowl, combine the tofu, mushrooms, scallions, cabbage, soy sauce, sesame oil and ginger. Season with salt and pepper.
In a large skillet, heat the oil over medium heat, add the vegetable mixture and cook until tender, about 5 minutes; transfer to a bowl and let cool.
Brush one wonton wrapper with water and place 1 heaping teaspoon filling in the center. Fold the wrapper into a triangle, squeeze out the air and press the edges together to seal, wetting the dough if necessary to make it stick. Pinch together the two corners of the folded edge. Repeat with the remaining wrappers.
Bring a pot of salted water to a boil. Working in batches, add the dumplings and boil until cooked through, about 3 minutes after the water returns to a boil. Remove with a slotted spoon and drain. Serve with dipping sauces.