Place the cabbage in a bowl and sprinkle with the salt. Set aside for 5 to 10 minutes.
Squeeze the liquid from the cabbage and transfer to another mixing bowl. Combine the cabbage with remaining filling ingredients. Work the mixture together with your hands until fully combined.
Clear a large space on the counter. Set a small bowl of water, the bowl of filling, and a parchment-lined baking sheet nearby. Open the package of dumpling wrappers and arrange a few on the workspace in front of you.
Place a heaping teaspoon of filling on each dumpling wrapper. Don’t overfill. Brush the edges of the wrapper with water. To fold, bring up the sides of the wrapper to meet in the middle. Press the middle closed then pleat to close from the middle to the corner, one side and then the other. You can also just fold the dumpling in half and press to close with a fork.
For purses, bring the 4 corners up together, over the filling, and press to close. (This works best with square wrappers.)
As you finish each dumpling, line it up on the parchment-lined baking sheet. Cook or freeze the dumplings.
To cook, add a tablespoon of oil to a skillet and warm over high heat. When the oil is shimmering, arrange the dumplings in the pan as close as they'll fit without touching. Cook until the bottoms have turned brown and golden. Place lid over the pan. Sneak it open just a little and pour 1/4 cup water in the pan — the water will immediately sizzle and begin to steam. Cover the pan and lower the heat.
Cook the dumplings for 3 to 5 minutes if fresh, or 6 to 8 minutes if frozen. When done, the wrappers will appear translucent and noodle-like; the filling will be opaque and warmed through. Cut one open to check. Transfer the cooked dumplings to serve.