Press the tofu to remove excess moisture and then crumble it into a large mixing bowl.
Roughly chop the shelled edamame, shredded cabbage, grated carrot, and green onions, and add them to the bowl.
Mince the garlic and grate the ginger; add them into the mixture along with a light drizzle of soy sauce and a pinch of salt and pepper.
Gently combine the ingredients until the filling is well mixed.
Place a dumpling wrapper on a clean surface and spoon a small amount of filling (about 1-2 tablespoons) into the center. Moisten the edges with a little water, fold the wrapper over the filling, and press to seal. Repeat with the remaining wrappers and filling.
Heat the sesame oil in a non-stick skillet over medium heat. Arrange the dumplings in the skillet without overcrowding.
Fry the dumplings for about 2-3 minutes until the bottoms are golden and crispy.
Add a splash of water to the skillet (about 2-3 tablespoons), cover immediately, and let the dumplings steam for another 3-4 minutes until the filling is cooked through.
Remove the lid and allow any remaining water to evaporate, crisping up the bottom again for an extra minute.
For the dipping sauce, combine soy sauce and rice vinegar in a small bowl. Serve the dumplings hot with the dipping sauce on the side.