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Preheat broiler. Line a large shallow baking pan with foil and lightly oil.
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Toss onion and Broccolini in a bowl with vegetable oil, 2 tablespoons teriyaki sauce, and 3/4 teaspoon salt.
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Pat tofu dry and arrange down middle of pan. Sprinkle with 3/4 teaspoon each of salt and pepper, then brush with 1 1/2 tablespoons teriyaki sauce. Spread vegetable mixture alongside tofu.
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Broil 3 to 4 inches from heat until tofu is pale golden and vegetables are lightly charred in spots, about 7 minutes. Turn tofu over and brush with 1 1/2 tablespoons teriyaki sauce, then stir vegetables. Broil until tofu is pale golden and vegetables are crisp-tender, about 7 minutes more.
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Meanwhile, whisk together water, cornstarch, and 2 tablespoons teriyaki sauce in a small heavy saucepan, then simmer, whisking occasionally, 2 minutes. Remove from heat and stir in lemon juice, sesame oil, and, if desired, additional teriyaki sauce. Serve sauce over tofu, vegetables, and rice.
