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.
MAKE AHEAD The lemon-ginger mixture can be frozen for up to 1 week.
Cooking Club Tip: Peel 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.