Cucumber with lemon butter is the cucumber sandwich with its single quiet note given a sharp partner, and the partner is in the butter rather than the filling. Plain cucumber on bread is cool and almost neutral; the move here is to fold lemon zest into the butter so the seasoning is built into the seal instead of laid on top. That is the whole identity. The cucumber stays restrained and watery-fresh, and the citrus does the work of waking it up, a bright acidic edge worked through the one structural fat the sandwich has, so every bite carries the lemon whether or not it carries much else.
The craft is in the butter and the drain. The butter is softened and beaten with finely grated zest, no juice, because juice loosens the butter and water is the one thing this sandwich cannot afford near the bread. Zest alone keeps the fat firm enough to spread to the edges and seal the crumb, and it distributes the citrus evenly so the lemon is a steady note rather than a sour patch. The cucumber is sliced thin, salted, and patted dry as in any cucumber sandwich, because the lemon butter sharpens the filling but does nothing to stop it weeping. Soft white bread is the carrier, the lemon butter spread firmly on both faces so the acid meets the cucumber from both sides, and the sandwich is cut and eaten soon after, since neither the cucumber nor the brightness of fresh zest holds for long.
The variations keep the flavoured-butter idea and swap the aromatic. Mint or dill worked into the butter instead of or alongside the lemon for a herb register; a little white pepper for heat under the citrus; cream cheese carrying the zest in place of butter for a richer body. The plain unflavoured cucumber finger is the restrained parent of all of these and, like each of them, deserves its own article rather than being crowded in here.