The roast lamb sandwich is the cold-lamb baseline of the roast cluster, and what it is built around is the gap that mint sauce is expected to fill. Cold roast lamb is a strong, faintly sweet meat with a fat that firms and waxes as it cools, so on its own between bread it sits heavy and one-note, a single rich register the whole way down. The expected counter is a sharp green mint sauce, vinegar-edged and barely sweet, cutting straight across the firmed fat with acid the way horseradish cuts beef and apple cuts pork. That counter is so closely linked to this meat that a plain lamb sandwich tastes, to most people, like a lamb sandwich missing its sauce. The lamb supplies the body; the question of the counter is the whole sandwich.
The craft is in the cut and the moisture. Cold roast lamb is at its best sliced thin and against the grain, because a thick cold slice is chewy and pushes the waxed fat forward, while a thin one stays tender and folds to the loaf. Off the joint the meat has lost the heat and the running juices it had on the plate, so the build has to put lubrication back: a measured smear of mint sauce, butter spread to the edges, or a thin film of the lamb's own dripping returned to the bread. The bread needs real structure for a dense filling, a sturdy white or a sliced bloomer rather than anything soft, and it is buttered partly to bridge the salt of the lamb to the wheat and partly to seal the crumb against a wet sauce that otherwise soaks through to a sour patch within minutes. Lamb is popular in Wales, where the joint is good enough that the discipline is to slice it well and not bury it.
The variations are mostly a question of which counter the lamb takes and a few related forms. Sharp vinegar mint sauce keeps it cleanest and is the default; mint jelly turns the same idea sweeter and set so it spreads without bleeding; a stripe of redcurrant pushes it toward the fruited register lamb also carries well. The plain salted slice without any sauce is the thinnest reading of all, and a hot lamb roll run with gravy belongs to a different tradition entirely. Each deserves its own article rather than being crowded in here.