In a large mixing bowl, combine ground beef and ground chicken.
Add minced garlic, grated ginger, and chopped green onions to the meat mixture.
Pour in soy sauce, sesame oil, salt, and black pepper. Mix thoroughly until all ingredients are well incorporated.
Place one dumpling wrapper on a clean surface. Spoon about 1 tablespoon of the meat filling into the center of the wrapper.
Dip your finger into the water and moisten the edges of the wrapper.
Fold the wrapper in half over the filling to form a half-moon shape. Press the edges together firmly to seal, making sure there are no air pockets.
Repeat the filling and folding process until all dumplings are assembled.
To cook by steaming: Place dumplings in a steamer lined with parchment paper or cabbage leaves to prevent sticking. Steam over boiling water for 10-12 minutes until filling is cooked through.
To cook by boiling: Bring a large pot of water to a boil. Add dumplings in batches and cook for 6-8 minutes or until they float to the surface and are cooked through. Remove with a slotted spoon.
To cook by pan-frying: Heat vegetable oil in a non-stick skillet over medium heat. Place dumplings in the skillet and cook for 2-3 minutes until bottoms are golden brown. Add 1/4 cup water to the skillet, cover, and steam for 5-6 minutes until water evaporates and dumplings are cooked through.
Serve hot with soy sauce or your favorite dipping sauce.