Cut the cucumber in half. Slice one half into circles and set aside in the refrigerator. Peel the other half, slice in half lengthwise, scrape out the seeds, and cut into chunks. Place the chunks of cucumber, mint, basil, and honey in a food processor or blender, add the vodka and blend until as smooth as possible; let the motor run for at least 30 seconds. Refrigerate the puree for at least 1 hour, until chilled.
Pour in the chilled puree into a pitcher, then pour in the Prosecco. Serve in the glass of your choice over ice, with the cucumber slices and additional sprigs of fresh herbs for garnish.
Notes
Serve in rocks glasses for the perfect presentation.