Place sugar and vinegar in a small bowl. Stir until sugar dissolves. Add carrot. Stir to combine.
Make Chilli Dressing: Place garlic, chilli, sugar, fish sauce, lime juice and 1/2 cup water in a small bowl. Stir until sugar dissolves.
Cook noodles following packet directions. Drain. Refresh under cold water. Drain.
Place chicken, lemongrass, garlic, fish sauce, extra sugar and white pepper in a glass or ceramic bowl. Toss to coat.
Heat a wok over medium-high heat. Add oil. Swirl to coat. Stir-fry chicken, in 2 batches, for 2 to 3 minutes, or until browned and cooked through. Transfer to a bowl. Cover to keep warm.
Drain carrot mixture. Divide noodles among serving bowls. Top with lettuce, cucumber, chicken, carrot, coriander and mint. Sprinkle with fried shallots and peanuts. Drizzle with Chilli Dressing. Serve.