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In a large bowl, combine the chicken, garlic, ginger, cilantro, tamari, sesame oil, 1 teaspoon salt, and the white pepper. Add the egg, sprinkle on the breadcrumbs, and mix thoroughly with your hands. Form 18 to 24 meatballs, each about 1-1/2 inches in diameter. Set aside.
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Place the noodles in a large heatproof bowl. Heat enough boiling water to cover. Let the noodles sit in the hot water, agitating occasionally, until they're tender, 10 to 12 minutes. Drain, rinse with cold water to stop the cooking, drain again, drizzle with a little oil, and toss to coat to prevent the noodles from sticking.
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Meanwhile, in a large skillet over medium-high heat, warm the oil. Working in batches to avoid crowding, cook the meatballs, turning often, until golden brown all over, 5 to 6 minutes. Transfer to a plate.
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Bring the chicken broth to a simmer in a large pot. Add the carrots, meatballs, and the unpeeled ginger slices and simmer for 5 minutes. Add the bok choy and simmer until all of the vegetables are fork tender but still bright in color, about 5 minutes longer. Season to taste with salt. Discard the unpeeled ginger slices.
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Divide the noodles among 4 bowls, ladle the meatball soup on top, garnish with cilantro leaves and green onions, and serve.