To prepare the glass noodles, in a bowl, cover the dried glass noodles with hot water and soak for about 2 minutes until softened, then drain well and cut into shorter segments.
To stir-fry the filling, in a pan over high heat, add the cooking oil. Add the carrot, cabbage, snow peas, and shiitake mushrooms, then stir-fry for about 2 minutes until slightly softened. Add the garlic and stir-fry for about 10 seconds until fragrant. Add the glass noodles along with vegetarian oyster sauce, hoisin sauce, and white pepper, then stir-fry until everything is evenly combined and excess moisture has cooked off. Turn off the heat and let the filling cool slightly.
To wrap the spring rolls, on a flat surface, place a spring roll wrapper in a diamond shape with one corner pointing toward you. Add about 2 tablespoons of filling near the bottom corner, fold the bottom corner over the filling, fold in both sides, and roll tightly. Brush the top corner with the egg to seal, then repeat with the remaining wrappers.
To fry the spring rolls, in a pot over high heat, add the cooking oil and heat to 350°F (177°C). Carefully add the spring rolls and fry for about 4 minutes, turning as needed, until golden brown and crispy. Remove and place on a rack to drain.