Slice ends off cucumber, then slice crosswise into three 2-inch sections. Stand one section at a time on its base; slice sides from top to bottom, squaring cucumber; reserve peelings. With apple corer, hollow out centers; leave 1/2 inch at bottom so sake will not leak out.
Cut cucumber peelings into matchsticks; dip into sake, and sprinkle with salt to bring out the sake's sweetness.
Cold sake, the Japanese rice wine, is sometimes served in a boxy cup and garnished with a cucumber slice. Our food editors took this one pretty step further by making the cups themselves out of fragrant cucumbers. Japanese cucumbers, with fewer seeds and less pulp, make the cleanest cups.
Tips: The chilled sake is not meant to be sipped, but downed in a single shot. If sake is not your drink, you can substitute vodka or tequila.