To soften the noodles, in a bowl, soak the glass noodles in the hot water for 5 minutes until softened. Drain well and cut into 2-inch lengths.
To make the dipping sauce, in a bowl, whisk together the soy sauce, Chinkiang vinegar, and chili oil. Set aside.
To cook the filling, in a pan over high heat, add the cooking oil. Stir-fry the carrot, cabbage, shiitake mushrooms, and garlic for about 2 minutes until aromatic and slightly softened. Add the noodles and mix well. Add the vegetarian oyster sauce and white pepper and stir-fry until evenly combined. Remove from heat and cool for 5 minutes.
To assemble, place a spring roll wrapper in a diamond position. Add about 3 tablespoons of filling to the center. Fold the bottom corner over the filling, fold in both sides, and roll upward tightly. Brush the top corner with beaten egg to seal. Repeat.
To pan-fry, in a pan over medium-high heat, add the cooking oil. Pan-fry the pockets for about 2 minutes per side until golden brown and crispy. Serve hot with dipping sauce.