For the ginger vodka: Place the ginger in an airtight container. Add a few ounces of the vodka and muddle for several minutes. Add the rest of the vodka, cover and let sit for at least 24 hours, muddling the ginger again every few hours. (Makes 18 ounces.)
For the ginger simple syrup: Heat the sugar and water in a small, heavy-bottomed saucepan over medium heat. When the sugar is completely dissolved, about 5 minutes, stir in the ginger and remove from heat. Let the ginger steep until the syrup is completely cool. Remove the ginger and transfer the syrup to an airtight container. Refrigerate until ready to use. (Makes about 12 ounces.)
For the ginger spritz: Pour 4 ounces ginger vodka, 2 ounces ginger simple syrup and the lime juice into a cocktail shaker. Add ice, cover, and shake until fully chilled.
Strain into two glasses and top each with 2 ounces club soda. Garnish with the candied ginger skewers.