In a large bowl, combine the shrimp, cabbage, carrots, green onions, garlic, ginger, soy sauce, sesame oil, cornstarch, salt, and pepper. Mix well until all ingredients are evenly 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 to seal. Repeat with the remaining wrappers and filling.
Heat a large skillet over medium-high heat and add a tablespoon of cooking oil. Once the oil is hot, add the dumplings in a single layer, making sure not to overcrowd the pan. Cook for about 2-3 minutes, or until the bottoms are golden brown.
Add 1/4 cup of water to the skillet and immediately cover with a lid. Steam the dumplings for about 5-6 minutes, or until the water has evaporated and the dumplings are cooked through. Remove the lid and let them cook for an additional minute to crisp up the bottoms.
Serve the dumplings hot with soy sauce or your favorite dipping sauce.