The cucumber sandwich is defined by everything taken out of it. Paper-thin cucumber on crustless white bread with salted butter, and nothing else: no second vegetable, no herb, no sauce. It is the canonical afternoon-tea finger, and its whole character is restraint pushed to the edge of austerity. The thinness is not delicacy for its own sake; it is the point. A thick cucumber slice is watery and assertive, but sliced to translucence it becomes a cool, faint, almost neutral note, and the sandwich is an argument that a single quiet ingredient, handled with discipline, is enough.
The craft is moisture control and proportion, and there is little else to hide behind. Cucumber is mostly water, so the slices are cut wafer-thin, salted, and left to drain, then patted dry, because any water left in them weeps into the bread and the sandwich is ruined by the time it reaches the table. Salted butter is structural, not a topping: spread firmly to the very edges, it waterproofs the crumb against the cucumber and its salt is the only seasoning the sandwich gets, the single sharp counter to an otherwise mild filling. The bread is soft white sliced thin, the crusts cut away after assembly so nothing with chew resists a filling chosen for its lack of resistance, and the sandwich is cut into fingers or triangles small enough to eat in two bites without a plate. It is made close to when it is served, because even a well-drained cucumber sandwich holds its brief freshness only a short while and goes limp if it waits.
The variations are the single additions that each earn a different name. Cream cheese for a richer body and a mortar that fixes the slip; dill or mint for one cool herb; smoked salmon to turn it into something heavier. Each is its own pairing rather than a tweak to this one, and each deserves its own article rather than being crowded in here.