1. Simmer peppercorns, vinegar, and sugar in a small saucepan over low heat until reduced by half, about 18 minutes. Stir in ketchup, then strain. Chill until ready to use.
2. Drop noodles into a bowl of very hot water. Let sit, stirring occasionally, until soft and clear, about 4 minutes. Drain, cut up a bit with scissors, and set aside.
3. Heat sesame oil and 1 1/2 tsp. vegetable oil in a large frying pan over medium-high heat. Add cabbage, carrots, mushrooms, salt, and pepper and cook until wilted and starting to brown. Add reserved noodles and stir to combine.
4. Preheat oven to 450°. Lay 1 egg roll wrapper on a work surface so a corner is facing you. Spoon about 1/4 cup filling just below center. Brush top corner with a bit of water. Lift bottom corner up and over filling, then fold in sides and roll to top corner to seal. Set finished egg roll on a baking sheet lined with parchment paper. Make more egg rolls the same way until filling is gone.
5. Brush rolls on both sides with remaining 1 tbsp. vegetable oil, turning to coat. Bake rolls, turning occasionally, until browned and crisp, 17 to 20 minutes. Cut in half and serve with sauce.
*Find Chinese red rice vinegar and wood ear mushrooms at Asian grocery stores and bean threads in the Asian-foods aisle of well-stocked supermarkets.
Make ahead: Up to 4 hours, chilled; reheat on a baking sheet in a 350° oven 10 to 15 minutes.
Note: Nutritional analysis is per serving.