Prepare the Hot Chinese Mustard Sauce. In a bowl, combine all the ingredients except the oil. Stir until well blended. Slowly whisk in the peanut oil until emulsified. Refrigerate, covered until needed.
Prepare the aromatic mixture. In a food processor, combine the ginger, garlic, green onions, and red pepper flakes. Turn on machine, slowly add oil, and process mixture to a puree. Saute mixture for 1 to 2 minutes. Season with salt, pepper, and sugar. Set aside.
Prepare the filling. In a wok, stir-fry each vegetable separately in peanut oil. Season with salt and pepper. Drain and transfer to a mixing bowl. Stir in noodles, mushroom soy sauce, chile paste, sesame oil, and cilantro. Season with salt and pepper. Set aside to cool completely. When cool, blend the filling mixture with the aromatic mixture. When ready to form the spring rolls, squeeze out all the excess liquid.
Make the spring rolls. Place about 1/2 cup of prepared filling per spring roll wrapper. Roll and seal edges with the egg wash. Repeat the process until all filling is used.
In a deep pot, with about 4 cups peanut oil heated to 350 degrees F., deep-fry in batches of 4 until golden, about 2 to 3 minutes. Drain on paper towels. Trim off the ends and slice diagonally. Drizzle top with Hot Chinese Mustard Sauce. Serve immediately.
*RAW EGG WARNING
Food Network Kitchens suggest caution in consuming raw and lightly-cooked eggs due to the slight risk of Salmonella or other food-borne illness. To reduce this risk, we recommend you use only fresh, properly-refrigerated, clean, grade A or AA eggs with intact shells, and avoid contact between the yolks or whites and the shell.