Whisk coconut milk, lime juice, fish sauce, shallot, and brown sugar in a small bowl until smooth. Gradually add oil, whisking constantly until emulsified; set aside.
Cook noodles in a large saucepan of boiling salted water according to package instructions until tender (you want them to be a little more cooked than al dente since they are served cold). Drain; rinse under cold water, then drain again.
Toss noodles, shrimp, cucumber, watercress, and cilantro in a large bowl with three-quarters of reserved dressing to combine and evenly coat; season with salt. Top with chiles and peanuts and drizzle remaining dressing over.
Do Ahead: Dressing can be made 1 day ahead. Cover and chill.
