Place the chopped cabbage in a bowl and sprinkle with the salt.
Let the cabbage rest for 10 minutes to release moisture.
Squeeze the cabbage firmly to remove excess liquid, then transfer to a clean bowl.
Add the ground pork or chicken, green onions, ginger, garlic, soy sauce, sesame oil, rice vinegar, and black pepper.
Mix gently until just combined, avoiding overmixing.
Cover the filling and refrigerate for 15 minutes.
Place one dumpling wrapper in your palm.
Add 1 tablespoon of filling to the center of the wrapper.
Lightly moisten the edges of the wrapper with water.
Fold the wrapper in half and pleat or press firmly to seal.
Repeat until all filling is used.
Heat the oil in a large skillet over medium heat.
Arrange dumplings flat-side down in the pan without overcrowding.
Cook for 3 minutes until the bottoms are golden brown.
Carefully add the water to the pan and immediately cover.
Steam for 5 minutes until the filling is cooked through.
Remove the lid and cook 1 additional minute to crisp the bottoms.
Transfer dumplings to a serving plate and serve warm.