Place enough sugar on small plate to reach depth of 1/4 inch. Run lime wedge around rim of 4 Martini glasses to moisten. Dip rims into sugar on plate. Blend rum, lime juice, and 1/3 cup sugar in blender until sugar dissolves. Add 4 cups ice; blend until slushy and almost smooth, adding up to 1 cup more ice, if desired. Add 12 mint leaves and blend until small green flecks appear. Pour into prepared glasses. Garnish with mint sprigs.