Make the filling: Combine the ricotta, mozzarella, tomato paste, parmesan, garlic, oregano, basil, chili flakes, salt, and pepper in a bowl. Mix well until fully combined. Taste and adjust seasoning.
Rest the filling: Cover and refrigerate for 15 minutes. This firms it up and makes it much easier to work with.
Fill and seal: Place a gyoza wrapper in your palm. Add about 1 teaspoon of filling to the center — don't overfill. Brush the edges with beaten egg, fold in half, and pinch firmly to seal. Double-pinch the edges to make sure there are no gaps. Repeat with remaining wrappers.
Heat the pan: Heat the oil in a shallow non-stick or cast-iron pan over medium heat until shimmering.
Pan fry: Place the gyoza in the pan in a single layer, flat side down. Cook for 2–3 minutes until the bottoms are golden brown. Add ¼ cup (60ml) of water to the pan and immediately cover with a lid. Steam for 2–3 minutes until the wrappers are translucent and the filling is melted. Remove the lid and cook for another minute to crisp the bottoms back up.
Serve: Let them rest for a minute before serving — the filling will be very hot. Serve with a simple marinara or garlic butter sauce on the side for dipping.