In a saucepan, add the sugar, water, and corn syrup and cook over medium heat until sugar is dissolved. Add the beets and cook over medium-low heat for 10 minutes. Then remove from heat and let cool to room temperature. Strain the syrup into a glass jar or container and add half of the beets, saving the other half to infuse your vodka. Store the syrup in the refrigerator.

Add the other half of the beets to 6–8 ounces of vodka and let sit in an airtight container on the counter for a day or up to 7 if you want a more pronounced beet flavor. In a cocktail shaker filled with ice, add the infused vodka, St. Germain, beet syrup, lemon juice, and bitters and shake until cold. Strain into tall rocks glass filled with ice and top with soda water. Garnish with sliced lemon and a mint sprig.