Fill a cocktail shaker or small pitcher with ice. Add the vodka, ginger syrup, and lime juice. Cover and shake vigorously, or stir, until combined and chilled, about 30 seconds. (In general, by the time the shaker mists up the drink is ready.) Strain into a chilled cocktail glass. Add twist and serve.
Remove the outer peel of the lemon with a vegetable peeler, taking care not to include too much of the bitter white pith. Finely chop the lemon peel and ginger in a food processor.
Transfer the lemon-ginger mixture to a medium saucepan, add the sugar and water and bring to a boil. Reduce heat, and simmer, partially covered, for 15 minutes. Strain the mixture and cool. Cover and refrigerate for up to 1 week.
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