Cook the noodles in boiling water until tender to the bite. Drain and cool the noodles to room temperature.
Combine the arrowroot with enough stock to form a slurry. Bring the remaining stock to a boil. Add the slurry and stir until thickened. Cool to room temperature.
Combine the lime juice, vinegar, soy sauce, ginger, lime zest, and garlic. Whisk in the thickened stock and sesame oil. Stir in the cilantro.
Heat the vegetable oil in a small saute pan. Add the mushrooms and saute until almost cooked. Add the snow peas and continue to saute until the peas are bright green.
Toss together the noodles, snow peas, mushrooms, and vinaigrette. Serve the salad at room temperature.