To prepare the crab filling, in a bowl, combine the crab meat, salt, white pepper, sesame oil, and scallions. Mix gently until combined.
To prepare the rice paper, in another bowl, beat the egg. Place a sheet of rice paper on a flat surface and lightly brush both sides with the egg until softened and flexible.
To assemble the dumplings, place some crab filling near the bottom edge of the rice paper. Fold the bottom over the filling, fold in the sides, and roll tightly to form a dumpling. Repeat with the remaining wrappers and filling.
To pan-fry the dumplings, in a pan over medium heat, add the cooking oil. Carefully place the dumplings in the pan, leaving a little space between them. Pan-fry for about 3 minutes per side, or until golden brown and crispy.
To serve, transfer the dumplings to a serving plate and serve hot with soy sauce, vinegar, chili oil, or your favorite dipping sauce.