In a medium bowl, whisk sugar and ½ cup hot water until sugar is dissolved. Whisk in chile, lime juice, and fish sauce.

Make Ahead: Nuoc cham can be made 3 days ahead. Store in an airtight container and refrigerate.

In another medium bowl, whisk hoisin sauce, peanut butter, vinegar, and chili garlic paste until combined.

Make Ahead: Peanut sauce can be made 3 days ahead. Store in an airtight container and refrigerate.

In a medium pot over medium heat, cook sugar, vinegar, ½ teaspoon salt, and 2 tablespoons water, whisking frequently, until sugar is dissolved, about 2 minutes.

Add shrimp and bring to a simmer. Cook, stirring constantly, until shrimp turns pink and curls up, 2 to 3 minutes. Transfer shrimp mixture to a small bowl.

Wipe out pot and fill halfway with water; season with remaining 1 tablespoon salt. Bring to a boil and cook noodles, stirring occasionally, until tender, checking 2 minutes earlier than package directions. Drain, then rinse noodles under cold water.

Transfer noodles to a medium bowl. Add 6 tablespoons nuoc cham and toss to combine.

In a large bowl, toss lettuce, carrot, and cucumber with ½ cup nuoc cham.

Divide noodles among bowls. Nestle salad mixture alongside. Top with shrimp, avocado, herbs, and peanuts. Serve with peanut sauce alongside.

Make Ahead: Shrimp and noodles can be cooked 2 days ahead. Store separately in airtight containers and refrigerate.