Cut pineapple into chunks and add to blender or food processor with about 1/2 cup water.
Blend until pureed and no chunks remain.
Add 1/2 cup pineapple puree, 2 tablespoons basil simple syrup, 3 tablespoons vodka to a cocktail shaker with ice.
Shake until chilled, then add to glass.
Top with Topo Chico or club soda.
Garnish with sprig of basil or pineapple chunk.
Basil Simple Syrup
Add 1 cup water, 1 cup basil leaves and 1/2 cup sugar to saucepan.
Bring to a boil and stir until all sugar is dissolved.
Simmer for 1 minute, then remove from heat and let steep for about 30 minutes or until cooled.
After the syrup is cooled, strain into glass jar and discard basil leaves.
This will keep up to 2 weeks in the fridge.
Notes
If you prefer to make this by the pitcher instead of by the glass, add....2 cups pineapple puree, 8 tablespoons simple syrup, and 1/2 cup vodka to a 4 cup pitcher. If you chill your liquids first, you won't have to use the cocktail shaker or ice. Pour mixture evenly into glasses and top with Topo Chico or club soda. Variations:
Most simple syrups are 1 part water to 1 part sugar. I don't like my cocktails really sweet, so I cut the sugar in half.
Obviously, this is a personal preference, so I would start with a smaller amount and add more if you want it sweeter.
If you use pineapple juice in the blender instead of water, it will make the puree a bit sweeter.