Preheat oven to 350 degrees F (175 degrees C). Lightly grease a baking sheet.
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.
Puree the carrot with water, the ginger, the whites of the scallions, the sesame oil, sugar, soy sauce, and the vinegar.
Toss with the squash and the scallion greens. Season to taste and top with toasted sesame seeds.