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Preheat oven to 350 degrees F (175 degrees C). Lightly grease a baking sheet.
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Sprinkle some water (about 1 tablespoon) in each side of the squash, and place it with cut sides down on the prepared baking sheet, and bake 30 minutes or until a sharp knife can be inserted with only a little resistance. Remove squash from oven and set aside to cool; then scrape with a fork.
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Puree the carrot with water, the ginger, the whites of the scallions, the sesame oil, sugar, soy sauce, and the vinegar.
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Toss with the squash and the scallion greens. Season to taste and top with toasted sesame seeds.