For the ginger syrup: Place all syrup ingredients in a saucepan and simmer until sugar dissolves and the ginger is infused in the syrup, about 20 minutes. Strain and allow to cool fully before use.
For the drink: Crush the frozen cranberries in a large glass pitcher. Add the lime juice and their juiced wedges to the pitcher. Pour in the vodka, cranberry juice, and ginger syrup. Put ice in low-ball glasses and fill glasses about 2/3 full. Top off with club soda and garnish with a shaving of fresh ginger.
NotesNote: Can be made without soda for a martini.