2/3 cup hoisin sauce (see Cooks' notes:)
1/3 cup creamy peanut butter
1/3 cup water
3 tablespoons fresh lime juice
1 1/2 tablespoons soy sauce
3 ounces dried rice-stick noodles
2 tablespoons fresh lime juice, divided
1 1/2 teaspoons sugar
2 3/4 cups coarsely grated carrot (about 1 pound; use large holes on a box grater)
3 tablespoons finely chopped serrano or jalapeño chiles, including seeds
12 (8-inch) rice-paper rounds, plus more in case some tear
4 cups thinly sliced napa cabbage, coarsely chopped
1 cup finely chopped cocktail peanuts
1 cup fresh mint leaves
1 cup fresh basil leaves (Asian or Italian)