Prepare all ingredients by shredding the cooked chicken, cooking and cooling the vermicelli noodles, and julienning the carrots and cucumber. Wash and dry the lettuce, mint, and cilantro.
Fill a shallow dish with warm water. Dip one rice paper wrapper into the water for about 10-15 seconds until it becomes soft and pliable.
Lay the softened rice paper on a clean surface. In the center, layer a small amount of shredded chicken, vermicelli noodles, lettuce, carrots, cucumber, mint, and cilantro.
Fold the sides of the rice paper over the filling, then roll it tightly from the bottom up to enclose the filling completely. Repeat with the remaining wrappers and ingredients.
Serve the summer rolls with peanut sauce for dipping.