In a large mixing bowl, combine the ground beef and ground chicken.
Add the minced garlic, grated ginger, and chopped green onions to the meat mixture.
Stir in the soy sauce, sesame oil, salt, and ground black pepper until all ingredients are evenly mixed.
Place one dumpling wrapper on a clean surface. Spoon approximately 1 tablespoon of the meat filling into the center of the wrapper.
Moisten the edges of the dumpling wrapper with a little water using your finger.
Fold the wrapper in half to form a half-moon shape, pressing the edges together to seal tightly. You may pleat the edges for a decorative finish.
Repeat the filling and folding process until all the filling and wrappers are used.
To pan-fry: Heat 1 tablespoon of vegetable oil in a large non-stick skillet over medium-high heat. Place dumplings in the skillet, flat side down, without crowding.
Cook for 2-3 minutes or until the bottoms are golden brown.
Add 60 milliliters of water to the skillet and immediately cover with a lid. Reduce heat to medium and steam the dumplings for 6-7 minutes until the water evaporates and the dumplings are cooked through.
Remove the lid and cook for another 1-2 minutes to re-crisp the bottoms.
Alternatively, to steam dumplings: place dumplings in a steamer basket lined with parchment paper or cabbage leaves, steam over boiling water for 10-12 minutes until cooked through.
Serve the dumplings hot with soy sauce or your favorite dipping sauce.