Make the mushroom filling
In a wok or large skillet, heat the oil over medium-high heat until shimmering hot. Add the garlic and cook until fragrant, about 15 seconds. Add the mushrooms and cook, stirring occasionally, until soft and slightly browned, 4 to 5 minutes. Stir in the light soy sauce and carrot and cook until the carrot is soft, about 2 minutes. Add the bean sprouts and cook until they just start to wilt, about 1 minute. Remove from the heat, season with white pepper, and set aside.
Make the dipping sauce
In a small blender or food processor (or in a mortar), combine the basil, garlic, lime zest, lime juice, soy sauce, 2 tablespoons water, sugar, and chiles. Blend (or mash with a pestle) until smooth. Let sit for 5 minutes for the flavors to meld before serving. (Alternatively, finely chop the basil and whisk with the remaining ingredients until the sugar dissolves.)
Make the rolls
Rinse the lettuce leaves and pat dry. Remove and discard the center ribs, then slice the leaves crosswise in half to yield about 24 pieces, each roughly 3 inches long. Arrange the lettuce, herbs, avocado, and mushroom filling on your work surface.
Fill a shallow bowl with lukewarm water and lay a damp kitchen towel on your workspace. One at a time, dunk a rice paper into the water until softened, then lay it flat on the towel.
To assemble, place about ¼ cup of the mushroom filling one-third of the way up from the bottom of the rice paper in a straight line. Top with mint and basil leaves, avocado slices, and lettuce. Roll up like a burrito: fold up the bottom, tuck in the sides, then roll tightly to the end. The rice paper will seal itself. Repeat with the remaining wrappers.
Serve immediately with the dipping sauce.
