How to Make It
In a large bowl, toss the daikon, carrots, cucumber and papaya with the salt and 1/2 tablespoon of the sugar. Let stand, stirring occasionally, until the vegetables are wilted and have released a good amount of liquid, about 1 hour. Drain and squeeze out as much liquid as possible. Wipe out the bowl and return the vegetables to it.
Meanwhile, bring a large saucepan of water to a boil. Add the vermicelli, cover and remove from the heat; let stand until softened, 3 to 4 minutes. Drain and cool under cold water. Drain again, pressing out the excess water. Using scissors, cut the vermicelli into 4-inch lengths. Add the vermicelli to the vegetables, along with the chopped mint, basil and cilantro.
In a small bowl, combine the lime juice, fish sauce, vinegar and mirin with the remaining 2 tablespoons of sugar; stir to dissolve the sugar. Add the ginger and shallot and pour all but 2 teaspoons of the dressing over the vermicelli and vegetables. Let stand for 1 hour.
Drain the mixture, pressing to squeeze out as much liquid as possible. Add the yogurt and mayonnaise to the reserved 2 teaspoons of dressing and refrigerate.
Fill a large pie plate halfway with warm water. Working with 1 or 2 rice papers at a time, soak the wrappers in the water until pliable, about 2 minutes. Lay the wrappers on a work surface and blot any excess water. Spoon about 1/2 cup of the vegetables and vermicelli onto the bottom third of a wrapper and tightly roll into a 5-inch log, tucking in the ends as you roll. Transfer to a cutting board. Cover the summer roll with a damp paper towel and a sheet of plastic wrap. Repeat with the remaining wrappers and filling.
Cut each roll into 4 pieces and stand them cut side up on a platter. Spoon about 1/2 teaspoon of the creamy sauce onto each roll and serve right away.