Bring a large pot of water to a boil. Cook the noodles according to the package instructions. Drain, rinse with cold water if necessary, and drain again.
Meanwhile, in a medium bowl, mix the arrowroot powder, soy sauce, vinegar, and 1½ cups water until smooth. Set the slurry aside.
In a large sauté pan, place the ginger, garlic, scallions, mushrooms, carrots, and ¼ cup water. Cover and cook over medium heat until the vegetables are halfway cooked, about5 minutes.
Add the broccoli and bok choy to the vegetables. Stir the reserved slurry and add it to the pan. Cover and cook until all the vegetables are crisp-tender and the sauce is slightly thickened, about 5 minutes.
Add the noodles to the vegetables; toss until well combined. Heat over medium-low if necessary to warm all the ingredients and add a little water to loosen the sauce, if necessary. Garnish with cilantro and cashews (if using) and serve hot.