Soak the rice sticks in enough hot water (about 110ºF) to cover by 1 inch until they become pliable, about 15 minutes.
Bring 2 quarts water and 1 tablespoon of the soy sauce to a boil in a 4 1/2-quart saucepan. Add the shrimp and cook until just firm, 1 to 2 minutes. Using a spider, remove the cooked shrimp to a cutting board and cool for 3 minutes. When cool enough to handle, peel, coarsely chop, and set aside.
Return the water to a boil. Drain the rice stick noodles and add to the hot soy-water. Cook until tender, approximately 3 minutes.
Meanwhile, whisk the lime juice, remaining 1 tablespoon soy sauce, chili paste, and sugar together in a medium mixing bowl.
Drain the noodles in a colander. Add the noodles to the lime juice mixture, toss, and set aside while preparing the vegetables.
Toss the cucumber, carrot, cilantro, and mint or basil together in a small bowl. Transfer any unabsorbed liquid from the noodles to the cucumber mixture and toss to combine.
Cut the noodles into small 1- to 2-inch pieces with kitchen shears.
Fill a pie dish with 3 cups warm water. Dip 1 rice paper wrapper in the water for 10 seconds, then transfer to a cutting board until the wrapper is pliable and slightly tacky, approximately 1 minute.
Place 1/4 cup of the vegetable mixture on the bottom third of the wrapper nearest to you. Spoon 2 tablespoons of the chopped shrimp on top of the vegetables. Top with 1/4 cup noodles. Bring the bottom edge of the wrapper tightly over the filling, and then fold in the sides. Finish rolling from bottom to top until the entire wrapper is rolled, being careful not to tear the rice paper. Place on a parchment-lined half-sheet pan and cover with a damp tea towel. Repeat with the remaining wrappers until the filling is gone. Wrap each roll in a lettuce leaf and serve with Soy Ginger Dipping Sauce.