Combine broth, ginger and garlic in a Dutch oven; bring to a simmer. Simmer, partially covered, over medium-low heat for 15 minutes. Discard the ginger and garlic.
Meanwhile, heat oil in a large nonstick skillet over medium-high heat. Add mushrooms and crushed red pepper; cook, stirring often, until tender, 3 to 5 minutes. Add bok choy stems; cook, stirring often, until tender, 3 to 4 minutes.
Add the mushroom mixture to the broth. Add noodles, reduce heat to medium-low and simmer for 3 minutes. Add bok choy greens and tofu; simmer until heated through, about 2 minutes. Stir in carrots, vinegar to taste, soy sauce and sesame oil. Serve garnished with scallions.
Ingredient note: Chinese wheat noodles and rice sticks (dried rice noodles) are quick-cooking and can be found in the Asian-food section of your supermarket.
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