In a shaker, muddle all ingredients except the ginger syrup.
Add the syrup and ice and shake well.
Strain into a highball glass (or stemless Champagne flute).
Garnish with a rosemary sprig.
*Ginger syrup: Bring 1/2 cup water to a boil in a small saucepan over high heat. Add 1 cup sugar and stir until dissolved. Remove from heat and let cool completely. Place 1 2-inch piece of ginger (finely chopped) in a mesh strainer and pour mixture over top. Press ginger with a muddler or back of a spoon to remove as much liquid as possible.