Muddle the basil and simple syrup together. Stir in peach puree/nectar and sweet tea vodka and add ice. Garnish with a lemon slice.
for the Lemon Simple Syrup
Combine sugar, water, and lemon zest in a small saucepan. Bring to a boil over medium high heat, stirring frequently, until sugar is dissolved. Pour through a fine mesh sieve into a small bowl. Discard solids and allow syrup to cool completely. Stir in lemon juice and refrigerate for up to 1 month.