Bring a medium pot of well-salted water to a boil. Add the noodles and cook, stirring, until tender, about 8 minutes. Transfer to a colander and run under cold water to cool slightly. Drain well.
In a medium bowl, mix the chicken broth, oyster sauce, vinegar, and 2 tsp. of the sesame oil.
Heat the ginger and canola oil in a large skillet over medium-high heat until the ginger sizzles steadily for about 30 seconds. Add the bok choy and mushrooms, sprinkle with the remaining 2 tsp. sesame oil and 3/4 tsp. salt and cook, tossing after 1 minute, until the bok choy turns dark green and begins to soften, 3 to 5 minutes. Add the chicken broth mixture, tofu, and carrots and bring to a boil. Reduce to a simmer, cover, and cook until the carrots are soft and the tofu is heated through, 5 to 7 minutes.
Distribute the noodles among 4 bowls. Spoon the vegetables, tofu, and broth over the noodles. Sprinkle with the scallions and serve.
nutrition information (per serving): Calories (kcal): 540; Fat (g): 19; Fat Calories (kcal): 170; Saturated Fat (g): 2; Protein (g): 24; Monounsaturated Fat (g): 9; Carbohydrates (g): 71; Polyunsaturated Fat (g): 4.5; Sodium (mg): 820; Cholesterol (mg): 0; Fiber (g): 8.