Add the water to a pot and bring it to a boil. Add the rice noodles and cook according to the package instructions, stirring occasionally. Drain and rinse under cold water, then set aside.
Combine all the ingredients for the Hoisin-Peanut Sauce together in a small bowl. Whisk it to mix well. Transfer to a dip bowl and garnish with the peanuts.
In another small pot, bring some water to a boil. Cook the shrimp for about 1 minute, or until they are completely cooked. Drain, let cool, and slice in half lengthwise. Set aside. Divide the vermicelli, shrimp, lettuce, and carrot into 6 equal portions.
To assemble the summer rolls, dip one sheet of rice paper in a large bowl of water. Shake off the excess water and quickly transfer it to a clean, dry, flat working surface, such as a kitchen countertop or a chopping board. Place the rice noodles on the bottom part of the rice paper.
Add the sliced lettuce and carrots.
Place 3 shrimp halves on top.
Fold the bottom side of rice paper over the filling securely, then fold the left and right sides of the rice paper over the filing. Make sure the filling is secured tightly.
Continue to roll the summer roll over, as pictured below. Repeat this process until all the ingredients are used. Cut the summer rolls diagonally in the middle into halves, place them on a platter, and serve immediately with the Hoisin-Peanut Sauce.