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 simple syrup 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. Top each with 4 oz. Prosecco and a splash of club soda. Gently stir together; garnish each with 3 cucumber ribbons.