To make the simple syrup:
Combine water and sugar in a pot medium-high heat. Stir until dissolved and bring to a boil.
Lower heat to a simmer, add cardamom and ginger. Stir and simmer for 15-20 minutes. Let cool.
To fashion the cocktail:
In a cocktail shaker combine gin, vermouth, fresh lemon juice, cardamom simple syrup, a rosemary sprig, and bitters.
Shake well with ice, and strain into a martini glass.
Garnish with rosemary and a slice of crystalized ginger.