In a large shaker, muddle 1/4 cup of the lime juice with 2 tablespoons of the superfine sugar and the ginger. Fill the shaker with crushed ice, then add 1 1/4 cups of the lime juice and the gin; shake the drink vigorously.
Pour the remaining 2 tablespoons of lime juice and 2 tablespoons of sugar into 2 bowls. Dip the rims of 4 cocktail glasses first in the lime juice and then in the sugar. Strain the cocktail into the glasses; garnish with the lime rounds.