For the Ginger Simple Syrup, place ginger in center of a piece of cheesecloth. Tie tightly with long piece of string. Place ginger sachet, water and sugar in small saucepan on medium heat. Bring to boil. Reduce heat to low; simmer 3 minutes, stirring to dissolve sugar. Remove from heat. Cool slightly. Refrigerate overnight to blend flavor. Remove sachet. Store spiced syrup, covered, in the refrigerator. (Makes 8 ounces)
For the Carrot Ginger-tini, fill cocktail shaker two-thirds full with ice. Add carrot juice, vodka, Ginger Simple Syrup and orange-flavored liqueur; shake until well mixed and chilled. Strain into martini glass. Garnish with shaved carrot, if desired. Serve immediately.