Start by making a honey simple syrup. The ratio is 3 parts honey to 1 part hot water. I measured 1/3 cup of honey into a small bowl then filled the 1/3 cup one third of the way full with hot tap water. Pour the hot water over the honey, gently stirring to combine.
To a cocktail shaker, add 3/4 oz. honey simple syrup, arugula and lime juice. Using a muddler (or a wooden spoon), mash up the arugula several times. The arugula should look bruised, not pulverized. Add the gin, cap and shake to mix the flavors.
Fill a short tumbler with ice. Strain the mixture through a fine-mesh strainer (this is called double straining) into the tumbler. Garnish with a mint sprig.