Rehydrate the shiitake. Soak the dried mushrooms in hot water for 30 minutes, then squeeze dry, discard the stems, and mince.
Draw out the cabbage water. Mix the cabbage with 1/2 tsp salt, wait 10 minutes, then twist it in a clean towel to squeeze out the liquid.
Stir the filling. Mix the cabbage, mushrooms, tofu, garlic, ginger, green onion, potato, sesame oil, soy sauce, sugar, 1/4 tsp salt, and pepper until it clumps together.
Fold the dumplings. Spoon a heaping teaspoon of filling onto each wrapper, moisten the edge, fold in half, and pinch to seal. Keep finished dumplings covered.
Crisp the bottoms. Heat 1 tbsp oil in a nonstick skillet over medium high heat and fry 16 dumplings flat side down for 2 minutes until golden.
Steam through. Add 1/4 cup water, cover, lower the heat to medium, and steam 4 minutes. Uncover and cook 1 minute more until the water is gone and the bottoms are crisp.
Repeat and serve. Cook the second batch the same way and serve hot with a dipping sauce.