To rehydrate the glass noodles, in a bowl, add the glass noodles and pour in hot water. Soak for about 2 minutes until slightly softened, then drain and cut into shorter segments.
To make the filling, in a pan over high heat, add 1 tablespoon of cooking oil. Stir-fry the carrot, cabbage, snow peas, and beech mushrooms for about 2 minutes until slightly softened. Add the garlic and stir-fry for another 10 seconds until aromatic. Add the glass noodles, oyster sauce, hoisin sauce, and white pepper, then stir-fry until evenly combined. Remove from the heat and let the filling cool completely.
To prepare the rice paper, in a bowl, crack the egg and whisk until smooth. Place a sheet of rice paper on a flat surface and lightly brush both sides with the egg until softened. Let it sit for a few seconds until pliable.
To assemble the dumplings, place a portion of filling near the bottom center of the rice paper. Roll into a log shape, then twist the ends to create a spiral shape. Repeat with the remaining filling and rice paper.
To pan-fry the dumplings, in a pan over medium heat, add the cooking oil. Once hot, place the dumplings in a single layer and pan-fry for about 2 minutes per side until golden brown and crispy, turning as needed.