Combine all ingredients in a pot set on medium-high heat. Stir occasionally, allow the syrup to boil, and then reduce heat to a simmer. Reduce by one cup of liquid. Remove pot from heat and allow to cool to room temperature. Strain out the ginger and place in a bottle or container with a tightly fitting lid. The syrup will keep for several weeks in the refrigerator, but ginger flavors are best when served closest to preparation.
Muddle the mint leaves and ginger syrup in the base of a mixing glass or tin. Combine other ingredients except garnish in the glass or tin. Add ice and shake vigorously. Double strain into a chilled cocktail glass. Garnish with a sprig of fresh mint.