Heat oven to 400° F. In a 9-by-13-inch baking dish, stir together the sugar, soy sauce, and 3 tablespoons of the lime juice; add the chicken and turn to coat. Roast, skin-side down, until the sauce is simmering, about 10 minutes. Turn the chicken skin-side up and roast, basting occasionally, until cooked through, 15 to 20 minutes more.
Meanwhile, in a large bowl, toss the cabbage, carrots, scallions, and mint with the ginger, canola oil, sesame oil, the remaining 3 tablespoons of lime juice, and ½ teaspoon each salt and pepper. Serve with the chicken.