In a large bowl, combine ground beef, ground chicken, minced garlic, grated ginger, and chopped green onions.
Add soy sauce, oyster sauce, sesame oil, salt, and ground black pepper to the meat mixture. Mix thoroughly until all ingredients are well combined.
Place a dumpling wrapper on a clean surface. Spoon about 1 tablespoon of the meat filling into the center of the wrapper.
Dip your finger in water and moisten the edges of the wrapper.
Fold the wrapper in half to form a half-moon shape and press the edges together to seal tightly, making sure there are no air pockets. You may pleat the edges for a traditional look.
Repeat the filling and folding process with the remaining wrappers and filling.
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 the filling is cooked through.
To pan-fry (potstickers): Heat 2 tablespoons vegetable oil in a large non-stick skillet over medium heat. Arrange dumplings in the skillet and 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. Cook for 8-10 minutes until the water has evaporated and the dumplings are cooked through.
Remove dumplings from heat and serve hot with soy sauce or your favorite dipping sauce.