Bring a large pot of water to a boil. Add the noodles and cook as the label directs; drain and rinse with cold water.
Meanwhile, whisk the lime juice, fish sauce, brown sugar, garlic, chili sauce and 1/3 cup water in a medium bowl. Transfer 1/4 cup of the marinade to another bowl and toss with the shrimp. Toss another 1/4 cup marinade with the asparagus and mushrooms in a third bowl. Let the shrimp and vegetables marinate 10 minutes at room temperature. Toss the noodles with the remaining marinade.
Heat a grill or grill pan to medium-high. Grill the shrimp, asparagus and mushrooms until the shrimp is just cooked through and the asparagus is slightly tender, 2 to 3 minutes per side. Halve the mushrooms and cut the asparagus into pieces.
Divide the noodles among bowls and top with the shrimp, asparagus, mushrooms, carrot and cilantro.