Use sharp knife to make 2-3 short slashes in skin of each piece of chicken, taking care not to cut into meat. Combine salt, sugar, and spices in small bowl and mix thoroughly. Coat chicken pieces with spices, gently lifting skin to distribute spice rub underneath but leaving it attached to chicken. Transfer chicken skin side up to wire rack set over rimmed baking sheet, lightly tent with foil, and refrigerate 6-24 hours.
Secure skin of each breast piece with 2-3 toothpicks placed near edges of skin.
Adjust oven rack to middle position; heat oven to 425 degrees. Roast chicken until thickest part of smallest piece registers 140 degrees on instant read thermometer, 15-20 minutes. Increase oven temperature to 500 degrees and continue roasting until chicken is browned and crisp and thickest parts of breast pieces register 160 degrees, 5-8 minutes longer, removing pieces from oven and transferring to clean wire rack as they finish cooking. Continue to roast thighs and/or drumsticks, if using, until thickest part of meat registers 170-175 degrees, about 5 minutes longer. Remove from oven; transfer chicken to rack and let cool completely before refrigerating or serving.