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, cooking 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 and season with the white pepper. 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, the sugar, and chiles and blend (or mash with a pestle) until a smooth sauce forms. Let sit for 5 minutes for the flavors to meld before serving. (Or, 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 so you end up with 24 pieces that are each about 3 inches long. Set the lettuce, herbs, avocado, and the mushroom filling on your work surface.
Fill a shallow bowl with lukewarm water to soak the rice papers. Lay a damp kitchen towel on your workspace. One at a time, dunk a rice paper into the water to soften, then lay it flat on the damp towel.
For assembly, place about ¼ cup of the mushroom filling about one-third up from the bottom of the rice paper wrapper in a straight line. Top with a few leaves of mint and basil, then avocado and lettuce leaves. Roll it up like a burrito by pulling up the bottom, tucking in the sides, then rolling to the end. The rice paper will stick together, sealing it shut. Repeat with the remaining rice papers.
Serve immediately with the dipping sauce.
