For the filling: Toss cabbage with salt in colander, set over bowl and let stand until cabbage begins to wilt, about 20 minutes. Press cabbage gently with rubber spatula to squeeze out any excess moisture, then transfer to medium bowl. Add the remaining filling ingredients and mix thoroughly to combine. Cover with plastic wrap and refrigerate until mixture is cold, at least 30 minutes or up to 24 hours.
For the dumplings: Working with 4 wrappers at a time (keep the remaining wrappers covered with plastic wrap), fill, seal and shape the dumplings using a generous 1 tsp of the chilled filling per dumpling. Transfer dumplings to a baking sheet and repeat with the remaining wrappers and filling; you should have about 24 dumplings.
The dumplings can be wrapped tightly with plastic wrap and refrigerated for up to 1 day, or frozen for up to 1 month. Once frozen, do not thaw before cooking.
Brush 1 Tbsp oil over bottom of 12" nontsick skillet and arrange half of the dumplings with flat side facing down (overlap if necessary). Place skillet over med-high head and cook dumplings without moving until golden brown on bottom, about 5 minutes.
Reduce heat to low, add 1/2 cup of water and cover immediately. Continue to cook, covered, until most of the water is absorbed and wrappers are slightly translucent, about 10 minutes. Uncover, increase heat to med-high and continue to cook without stirring until the dumpling bottoms are well browned and crisp, 3-4 minutes more.