Add the cucumber and dill syrup to the base of a cocktail shaker and muddle to a puree. Add all other ingredients with ice to the shaker and shake vigorously. Fine strain into a chilled martini glass. Garnish with a slice of cucumber, sprig of dill, and a grind of pepper.
* To make dill syrup, combine a cup of sugar and half a cup of water in a small saucepan. Turn up to a medium heat and stir until the sugar is dissolved. Add a handful of dill and simmer for 10 minutes. Take off the heat and leave to cool for an hour or two. Strain the syrup into an empty bottle and keep in the fridge for up to 6 weeks.