Place the finely chopped cabbage in a large bowl and sprinkle with salt.
Let the cabbage sit for 10 minutes to release moisture.
Squeeze the cabbage firmly with clean hands or a kitchen towel to remove as much liquid as possible.
Transfer the squeezed cabbage to a clean bowl.
Add the green onions, garlic, ginger, soy sauce, sesame oil, rice vinegar, and black pepper.
Mix until the filling is evenly combined.
Place one dumpling wrapper on a clean surface and add about 1 tablespoon of filling to the center.
Dip your finger in water and lightly wet the edge of the wrapper.
Fold the wrapper in half and press the edges firmly to seal. Pleat the edges if desired.
Repeat with the remaining wrappers and filling.
Heat 1 tablespoon of cooking oil in a large skillet over medium heat.
Add dumplings in a single layer, leaving a little space between each one.
Cook for 2 to 3 minutes, until the bottoms are golden brown.
Carefully add 1/3 cup water to the skillet and cover immediately.
Steam for 5 to 7 minutes, until the wrappers are tender and the filling is cooked through.
Remove the lid and cook for 1 more minute to crisp the bottoms again.
Repeat with remaining dumplings if cooking in batches.
Serve hot with dipping sauce.