In a large bowl, combine ground beef, ground chicken, chopped green onions, grated ginger, minced garlic, 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 teaspoon of the meat filling into the center of the wrapper.
Dip your finger in water and moisten the edge of the wrapper. Fold the wrapper in half over the filling to create a half-moon shape. Press the edges firmly to seal, making pleats along the edge if desired to ensure a tight seal.
Repeat the filling and folding process with the remaining wrappers and filling.
Heat 1 tablespoon of vegetable oil in a large non-stick skillet over medium-high heat.
Place dumplings in the skillet in a single layer, making sure they do not touch. Cook for 2-3 minutes or until the bottoms are golden brown.
Carefully pour 1/3 cup of water into the skillet and immediately cover with a lid. Let the dumplings steam for 6-8 minutes or until the water has evaporated and the dumplings are cooked through.
Remove the lid and cook for an additional 1-2 minutes to crisp up the bottoms again.
Transfer dumplings to a serving plate and serve hot with soy sauce or your preferred dipping sauce.