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Preheat the oven to 400°F. Cut the broccoli into bite-size florets and cut the stem into 1-inch chunks.
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In a large pot, cook the soba according to package directions. Drain, rinse well with cold water, and set aside.
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In a small bowl, whisk together the miso, honey, rice vinegar, tamari, and sesame oil.
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Place the sweet potatoes on a large, rimmed baking sheet, drizzle with 1 tablespoon of the olive oil, season with salt and pepper, toss to coat, then roast for 20 minutes. Add the broccoli, drizzle with the remaining 1 tablespoon olive oil and about two-thirds of the miso sauce. Toss, season with more salt and pepper, and roast until the broccoli edges are lightly charred and the sweet potato is tender, 12 to 15 minutes longer, tossing once halfway through cooking. Remove from the oven and toss the vegetables to coat.
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Meanwhile, in a small saucepan over medium-low heat, warm the broth and ginger.
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In a large bowl, combine the soba with the warm broth and the remaining one-third of the miso sauce. Toss to combine and divide among 4 bowls. Top each bowl with one-fourth of the vegetables, garnish with some green onions and sesame seeds, and serve.