Cook the shrimp. In a pot of boiling water, cook the shrimp for 2-3 minutes until they turn pink and are cooked through. Remove from the heat and let them cool. Once cool, cut the shrimp in half lengthwise.
Prepare the dipping sauce. In a small bowl, whisk together the fish sauce, soy sauce, sugar, lime juice, and minced garlic until the sugar is dissolved. Set aside.
Assemble the summer rolls. Fill a shallow dish with warm water. Dip a rice paper wrapper in the water for a few seconds until it becomes soft and pliable. Place the wrapper on a clean, damp towel. On the bottom third of the wrapper, add shrimp and a sprinkle of each of the shredded carrots, lettuce, cilantro, mint leaves, and chopped peanuts. Fold the sides of the wrapper towards the center and then roll it tightly from the bottom up. Repeat with the remaining ingredients.
Serve and enjoy! Arrange the Thai shrimp summer roll on a plate and serve with the dipping sauce.