Soak the rice sticks in enough hot water to cover by 1 inch for 15 minutes.
Bring 1/2 gallon of water to a boil in a large pot set over high heat. Add 1 tablespoon soy sauce. Add the shrimp and cook until just firm, 1 to 2 minutes. Remove the cooked shrimp to a cutting board and cool for 3 minutes. When cool enough to handle, peel, devein, and coarsely chop. Set aside.
Return the water-soy sauce 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 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 warm water. Dip 1 rice paper wrapper into 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 1/3 of the wrapper nearest to you. Spoon 2 tablespoons chopped shrimp on top of the vegetables. Top with 1/4 cup noodles. Bring the bottom edge of the wrap tightly over the filling, and then fold in the 2 sides. Finish rolling from bottom to top until the entire wrapper is rolled. Be 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.
Add all of the ingredients to a lidded jar and shake well to combine. Serve as a dipping sauce for spring rolls.