Using a vegetable peeler, shave 12 long, thin ribbons from cucumber; set aside.
Cut a 6" piece from remaining cucumber and cut into small pieces.
Muddle in a cocktail shaker, then add gin and cooled simple syrup (be sure it's not hot!) and fill shaker with ice.
Cover and shake vigorously until outside of shaker is frosty, about 30 seconds.
Strain mixture into 4 ice-filled rocks glasses or large wine glasses. T
Top each with 4 oz. Prosecco and a splash of club soda.
Gently stir together; garnish each with 3 cucumber ribbons.