Poach the chicken breast in simmering water until the internal temperature reaches 74°C, then shred and season with sea salt and black pepper.
Cook the rice vermicelli noodles in boiling water according to package instructions, rinse with cold water, and drain well.
Prepare the vegetables by julienning the carrots and cucumber into thin matchsticks and tearing the romaine lettuce into large pieces.
In a small bowl, whisk together the peanut butter, tamari, rice vinegar, lime juice, grated ginger, minced garlic, sesame oil, and water until the sauce is smooth and pourable.
Briefly dip one rice paper wrapper into a shallow dish of warm water for about 5 seconds until it begins to soften, then lay it flat on a clean, damp surface.
Place a piece of lettuce in the lower third of the wrapper, then top with a portion of noodles, shredded chicken, carrots, cucumber, cilantro, and mint.
Fold the bottom of the wrapper over the filling, fold in the left and right sides, and roll upward tightly to seal the roll.
Repeat the process for the remaining wrappers and serve immediately with the peanut dipping sauce on the side.