In a large bowl, combine the shrimp, napa cabbage, carrots, green onions, garlic, ginger, soy sauce, sesame oil, and cornstarch. Mix until well combined.
Place a dumpling wrapper in the palm of your hand. Spoon about 1 tablespoon of the filling into the center of the wrapper. Moisten the edges with water, fold the wrapper in half, and pinch the edges to seal. Repeat with the remaining wrappers and filling.
Heat a large skillet over medium heat and add a tablespoon of cooking oil. Arrange the dumplings in the skillet and cook for 2-3 minutes until the bottoms are golden brown.
Add 1/4 cup of water to the skillet and cover immediately. Allow the dumplings to steam for about 5-7 minutes until cooked through. Remove the lid and let any remaining water evaporate.
Serve the dumplings hot with soy sauce or your favorite dipping sauce.