Making Pan-Fried Cabbage and Noodle Buns is simple when you follow these easy steps:

1. Prepare the Filling: In a large bowl, combine chopped cabbage, cooked noodles, soy sauce, sesame oil, garlic, ginger, green onions, and the beaten egg. Mix well and season with salt and pepper to taste.
2. Make the Dough: In a separate bowl, mix flour with water until a soft dough forms. Knead briefly, then cover with a damp cloth and let rest for 10 minutes.
3. Roll Out the Dough: Divide the rested dough into small balls. Roll each ball into a thin circle on a floured surface.
4. Fill the Buns: Place a spoonful of the filling mixture in the center of each circle. Carefully fold the dough over the filling and pinch to seal the edges securely.
5. Pan-Fry the Buns: Heat oil in a skillet over medium heat. Place the buns seam-side down in the skillet, and fry for about 3-5 minutes until golden brown. Flip and fry the other side until browned as well.
6. Cook Thoroughly: Add a splash of water to the skillet, cover, and let steam for another 5 minutes to ensure the filling is heated through.
7. Serve: Once cooked, remove from the skillet and keep warm.

Following these steps will yield delicious buns that are crispy on the outside and filled with flavor on the inside!