In a 3- to 4-quart pan over high heat, bring 1 1/2 to 2 quarts water to a boil. Meanwhile, rinse chicken; chop into 1/4-inch pieces. Add bean threads or angel hair pasta to boiling water and stir to separate; cook until barely tender to bite, 3 to 4 minutes. Stir in chicken and cook just until chicken is white in center of thickest part (cut to test) and noodles are tender, about 1 minute. Drain noodles and chicken; rinse with cold water and drain thoroughly.
In a bowl, combine noodle mixture with shrimp, red onion, pickled garlic, bell pepper, cilantro, and green onion. In a small bowl, mix liquid from pickled garlic, lime juice, sugar, and fish sauce to taste.
Just before serving, combine dressing with noodle mixture. Divide arugula equally among dinner plates, or arrange all on a large platter. Mound glass noodle salad on or next to greens. Drizzle any remaining dressing left in bowl over arugula.