Make the filling: Combine the pizza cheese, tomato paste, parmesan, garlic, oregano, basil, chili flakes, salt, and pepper in a bowl. Mix until evenly combined. Taste and adjust seasoning.
Rest the filling: Cover and refrigerate for 15 minutes. This is especially important without a binder — cold filling is much easier to work with.
Fill and seal: Place a gyoza wrapper in your palm. Add a small amount of filling (less than 1 teaspoon) to the center. Brush the edges with beaten egg, fold in half, and pinch very firmly to seal. Double-pinch the edges — there is no binder to hold things together so a good seal is critical. Work quickly with the filling cold.
Heat the oil: Pour the oil into a frying pan to about ½–¾ inch (1–2cm) depth. Heat to 175°C (350°F). Keep a thermometer handy if you have one — the temperature will drop when you add gyoza so let it recover between batches.
Shallow fry: Fry in batches of 4–6, turning every minute or so to get even color on all sides. Fry for 2–3 minutes total until deep golden brown all over. Remove and drain on paper towels.
Serve: Rest for a minute before eating — the cheese inside will be extremely hot. Serve with marinara or a simple tomato dipping sauce on the side.