Whisk the lime juice, fish sauce, and sugar together in a small bowl until the sugar has completely dissolved.

Bring 4 quarts of water to a boil in a large pot. Remove the pot from the heat and stir in the rice vermicelli and salt. Let the noodles sit, stirring occasionally, until they are tender and translucent but not mushy, about 10 minutes.

Drain the noodles thoroughly, transfer them to a medium bowl, and toss with 2 tablespoons of the prepared fish sauce mixture.

In a small bowl, toss the carrot, peanuts, and jalapeño with 1 tablespoon of the fish sauce mixture. In a separate bowl, toss the cucumber with the remaining fish sauce mixture.

Prepare your workstation by laying a clean, damp kitchen towel on the counter and filling a shallow dish with 1 inch of room-temperature water. Submerge a rice wrapper in the water until it is just pliable but not yet floppy, about 2 minutes, then lay it flat on the towel.

Scatter several basil and cilantro leaves over the center of the wrapper. Arrange a few cucumber sticks horizontally across the middle, leaving a clear border at the bottom and sides. Top with a spoonful of the carrot mixture and a small portion of noodles.

Fold the bottom of the wrapper up over the filling, tuck in the sides, and roll the wrapper into a tight cylinder. Place the roll on a lettuce leaf and cover with another damp towel to keep it from drying out while you repeat the process.

Serve immediately with peanut dipping sauce, wrapping the lettuce around the outside of each roll as you eat for a clean, handheld snack.