Tear 5 basil leaves into small pieces and add them to a cocktail shaker. and gently muddle them to release their oils.
Zest 1/2 of the lime into the cocktail shaker.
Squeeze the lime into the cocktail shaker.
Add 3/4 ounce of simple syrup to the cocktail shaker.
Muddle the basil leaves and let it sit for about 5 minutes.
Add 2 1/2 ounces of vodka to the cocktail shaker.
Fill the shaker with ice and shake well.
Double-strain into two chilled cocktail glasses (Place a Hawthorne or built-in shaker strainer over the shaker, hold a fine-mesh strainer over the serving glass, and pour the drink through both strainers at once into the glass).