Soak vermicelli in water 15 minutes; drain and squeeze out water. Bring a pot of water to a boil. Place vermicelli in sieve and lower into water; remove after 5 seconds. Cool and divide into 12 portions.
Soak one rice paper in lukewarm water until pliable, about 5 minutes. Work with only 1 paper at a time and blot dry before assembling.
To assemble: Place 1 paper on a clean kitchen towel, leaving 1-inch of the rice paper uncovered on the edge closest to you. Place 1 lettuce leaf on the sheet. Cover with 1 portion of vermicelli, and top with a portion of the cucumbers, carrots, chicken, scallions, and mint leaves. Fold the paper once over the filling, starting from the bottom, then fold in the sides and continue rolling tightly to the end. Keep rolls covered with a damp towel. Serve with Spicy Peanut Sauce.
Spicy Peanut Sauce:
In a medium saucepan, heat the oil over medium heat. Add the curry paste and stir-fry until fragrant, about 2 minutes. Add the shrimp paste and continue to stir until the shrimp paste is broken up and one shade darker, about 1 minute. Add the peanuts and stir, roasting until two shades darker but not burned, 8 to 10 minutes. Add the sugar and continue to cook until the sugar is dissolved and starts to caramelize, 1 to 2 minutes. Add the coconut milk, chicken broth, tamarind concentrate, and hoisin sauce. Reduce the heat to low and simmer the sauce until slightly thickened, about 20 to 30 minutes. The natural oils from the peanuts should have surfaced at this point. Once the sauce has reduced by about 1/3, turn off the heat and add the Thai basil and cilantro. Cover and let cool. Store in the refrigerator for up to 3 days.