In a large mixing bowl, combine ground beef and ground chicken.
Add finely chopped napa cabbage, green onions, minced garlic, and grated ginger to the meat mixture.
Pour in soy sauce, sesame oil, sugar, salt, and black pepper.
Mix all ingredients thoroughly until well combined.
Place one dumpling wrapper on a clean surface and spoon about 1 tablespoon of filling into the center.
Moisten the edges of the wrapper with a little water using your finger.
Fold the wrapper in half over the filling and pinch the edges together to seal tightly, creating a half-moon shape.
Repeat the filling and folding process until all filling and wrappers are used.
To steam: Place dumplings in a steamer basket lined with parchment paper or cabbage leaves to prevent sticking. Steam over boiling water for 10-12 minutes until filling is cooked through.
To pan-fry: Heat vegetable oil in a large non-stick skillet over medium heat. Arrange dumplings in the skillet without crowding. Cook for 2-3 minutes until the bottoms are golden brown.
Add 1/4 cup water to the skillet and immediately cover with a lid. Let dumplings steam for 8-10 minutes until water evaporates and filling is cooked.
Remove lid and cook for another 1-2 minutes to crisp the bottoms again.
Serve hot with soy sauce or your favorite dipping sauce.