Place the chicken breast in a small pot of simmering water seasoned with sea salt and black pepper; cook for 12-15 minutes until the internal temperature reaches 165°F.
Remove the chicken from the pot, let it cool for a few minutes, and then shred it into thin strips using two forks.
While the chicken cools, whisk together the peanut butter, tamari, lime juice, and sriracha in a small bowl until the dipping sauce is smooth and creamy.
Prepare the vegetables by julienning the cucumber and shredding the carrots into thin matchsticks.
Fill a shallow dish with warm water and dip one rice paper wrapper for 5-10 seconds until it becomes soft and pliable.
Lay the wrapper flat on a clean surface and place a portion of the shredded chicken, cucumber, carrots, cilantro, and mint in the center.
Fold the bottom of the wrapper over the filling, tuck in the sides tightly, and roll upward to seal the summer roll.
Repeat the assembly process for the remaining wrappers and serve immediately with the prepared peanut-lime dipping sauce.