Combine pork, green onions, soy sauce, pepper and egg in a mixing bowl and stir to combine. Place 1 heaping teaspoon of the pork mixture into the center of each gyoza wrapper. Dip your fingers in a small bowl of water and then run them along the outer edges of the wrapper to dampen. Pull the sides of the wrapper together, forming a half moon shape and pinch the edges together to fully enclose the pork. Gently press down to flatten the filled bottom of the filled gyoza so that it stands upright. As you finish the gyoza, place them on a parchment paper-lined baking sheet so that they are close together but not touching. (Note: At this point the gyoza can be frozen or refrigerated until cooking. If you refrigerate them, gently cover them with slightly dampened paper towels to prevent them from drying out.)
To cook the gyoza, heat vegetable oil over medium-high heat in a pan that has a tight-fitting lid. When the oil is hot, place the gyoza in the pan so that they are close together but not quite touching. (You can make these in two batches if all gyoza will not fit in one pan.) Cook the gyoza, uncovered until the bottoms begin to turn brown and crisp, about 4 minutes. Reduce the heat to medium and gently pour ⅓ cup water into the pan. Be cautious, as it may splatter as it hits the oil. Put the lid on the pan and allow the gyoza to steam until the pork is cooked through, 7-10 minutes.
Serve warm with honey-soy dipping sauce on the side.