In a saucepan, bring the 2 cups water to a boil. Add the shrimp and immediately remove the saucepan from the heat. Cover and poach until pink and opaque throughout, about 3 minutes. With a slotted spoon, transfer the shrimp to a bowl of ice water and let cool for 3 minutes. Drain and cut each shrimp in half lengthwise. Refrigerate until ready to use.
In a heatproof bowl, combine the noodles and boiling water and soak for 10 minutes. Drain and return the noodles to the bowl. Add the carrot, cucumber, cabbage, bean sprouts and cilantro. Toss gently to mix.
Place a double thickness of paper towels on a work surface. Fill a large, shallow baking dish with water. Place 1 rice-paper wrapper in the water and soak until pliable, about 30 seconds. Carefully transfer the wrapper to the paper towels and turn once to blot dry. Arrange 1/2 cup of the noodle mixture on the bottom half of the wrapper.
Fold the bottom edge toward the center and roll up the wrapper halfway, making sure to wrap tightly around the filling. Tuck 2 basil leaf halves along the inside crease of the half-rolled wrapper. Arrange 4 pieces of the shrimp, cut sides up, along the crease. Fold the right and left edges of the wrapper over the filling and finish rolling up. Repeat with the remaining wrappers, filling, basil and shrimp. Transfer the rolls to a plate and cover with dampened paper towels.
To make the sauce, combine the hoisin sauce, green onion, lime juice, fish sauce, peanuts, red pepper flakes and brown sugar in a small bowl. Stir until well-blended.
To serve, cut the rolls in half on the diagonal and place on small individual plates. Pool the sauce alongside each roll.