Slice cucumber into 1/16-inch rounds, toss with salt in a colander and stand for 15 minutes. Spread on clean towels and blot thoroughly on both sides.
Beat butter with chives, lemon zest and white pepper until spreadable. Lay out the bread and spread one face of every slice evenly with the butter.
Arrange cucumber over eight buttered slices in one thin overlapping layer, trimming stray edges rather than folding them into thick spots. Close with the other eight slices, buttered faces inward.
Put the sandwiches in a covered container and refrigerate for 15 minutes to firm the butter slightly. Keep the bread flat without weighting it heavily.
Trim the crusts with a sharp serrated knife and cut each sandwich into three equal fingers, wiping the knife between cuts. Serve 24 fingers, two per serving, or refrigerate covered until serving the same day.