Whisk 1/4 cup rice vinegar with 2 teaspoons sugar in a medium bowl until sugar dissolves. Add chicken and carrot, season with salt to taste, set aside.
Fill a large bowl with warm water. Working with 2 sheets of rice paper at a time (keep others covered with a barely damp cloth to prevent curling), immerse papers in the warm water until slightly softened (about 15 seconds). Remove and spread out on a clean surface or cutting board. Pat dry with a towel to remove excess water. Lay 2 pieces of lettuce over the bottom third of the rice paper, leaving about 1/2-inch clear on edges. Place about 1/3 cup chicken and carrot mixture on the lettuce, top with 4 to 5 pieces cucumber, scallion, several mint leaves and about 1 heaping tablespoon of rice. Roll up the paper halfway into a cylinder. Fold both edges in to tuck while continuing to roll the paper to seal. Place the rolls on a plate covered with a damp towel so they stay moist as you prepare the remaining rolls. Cut rolls in half and wrap in plastic wrap or place in an air-tight container.
In a separate bowl, whisk together the peanut butter with the remaining 1 to 2 tablespoon rice vinegar, 2 teaspoons sugar, 2 tablespoons soy sauce and water until smooth. Put sauce in sealed containers.
Tuck rolls and sauce into a lunch sack with a cold pack and send off to school.
Note: Edible rice papers are sold in Asian food aisle in supermarkets or Asian markets.
Time Saving Tip: Use leftover rice from Chinese takeaway to make these quicker. Make them the night before or prep all ingredients including chicken and carrot mixture so in the morning you just need to roll.