How to Make It
In a small saucepan, combine the sliced ginger with the water and sugar and bring to a simmer over moderate heat. Simmer for 5 minutes, then let the ginger syrup cool slightly. Transfer to a blender and puree. Strain the syrup into a large, shallow glass dish. Stir in the lemon zest and juice. Cover and freeze the lemon-ginger syrup until firm, at least 4 hours.
Let the frozen lemon-ginger mixture stand at room temperature for 5 minutes. Chop the mixture into large pieces. Put half of the pieces in a blender, add 4 cups of crushed ice and 1 cup of vodka and blend until slushy. Pour the vodka cocktail into 4 glasses, garnish each with a mint sprig and serve. Repeat with the remaining frozen lemon mixture, ice, vodka and mint sprigs. Serve at once.
Peel the ginger's thin and delicate skin by scraping down the root with a metal teaspoon. A spoon maneuvers around the nubs more easily than a pairing knife.