Peel cucumber and puree in blender. Strain through a mesh strainer, pushing on cucumber pulp with a spatula or spoon to get as much juice out as you can. Reserve cucumber juice and throw out the leftover pulp. Set aside.
Combine vodka, cucumber juice, mint simple syrup*, and lemon juice in cocktail shaker filled with ice. Shake well. Strain into tall glass filled with ice. Top with lemonade (and a lithe club soda if you want it bubbly) or San Pellegrino Limonata.