Make the basil simple syrup: Add basil, sugar, and ½ cup water to a small Stainless Clad Saucier or Saucepan. Heat over medium and cook, stirring occasionally, until sugar has dissolved, 1–2 minutes. Remove from heat and let cool for 15 minutes. Strain through a fine-mesh sieve set over a Medium Bowl. Discard solids and refrigerate syrup until cold. You should have about ¾ cup syrup, enough to make about 12 cocktails.

Make the tomato juice: Slice tomatoes in half using a Utility Knife. Place cut side on a coarse grater and grate tomatoes into a Medium Bowl, discard skins. Add salt and stir to combine. Let sit for 5 minutes, then strain through a fine-mesh sieve set over another Bowl. Discard solids and refrigerate juice until ready to use. You should have about 1 cup juice, enough to make about 4 cocktails.

Make the cocktail: Add basil leaves and basil simple syrup to a cocktail shaker and muddle gently, releasing oils from basil. Fill shaker about halfway with ice. Add tomato juice, vodka, lemon juice, and balsamic vinegar. Shake vigorously until outside of shaker is very cold, 15–20 seconds. Fill a White Wine Glass with ice and strain cocktail into Glass. Top cocktail with sparkling water and a drizzle of olive oil (if using) and skewer any desired garnishes and add to Glass.