In a medium bowl, combine the shrimp, three-quarters of the garlic and the sesame oil; refrigerate for 30 minutes or up to overnight. In a small bowl, combine 1/4 cup warm water, the fish sauce, lime juice, honey and remaining garlic.
Preheat a large skillet over medium-high heat. Add the shrimp and cook, stirring, until opaque, about 3 minutes.
Fill a 9-inch round cake pan with an inch of warm water. Working with 1 at a time, dip the rice paper wrappers in the water, turning once, until softened, about 6 seconds, and place on a paper-towel-lined work surface. Place one-eighth of the cucumber and carrots across the center of the wrapper. Arrange 3 shrimp across the vegetables and top with 3 mint leaves. Fold up the bottom edge over the filling, then fold in the 2 sides and roll closed. Repeat with the remaining wrappers. Place each roll on a lettuce leaf and serve with the dipping sauce.