The roast beef sandwich is the cold-rare-beef baseline against which every other roast sandwich is measured, and what defines it is horseradish. Cold roast beef on bread is pleasant and slightly flat: a lean, faintly sweet, close-grained meat that on its own reads as one quiet savoury note across a whole sandwich. A smear of hot horseradish, sharp enough to catch the back of the nose, does the job mustard does for ham. It lifts the beef, cuts the residual fat, and supplies the heat the meat lacks, and it is the part that tells you this is a beef sandwich rather than cold meat between slices. The beef carries the body; the horseradish carries the point.
The craft is in the slicing and the moisture. Roast beef taken cold from the joint is at its best cut thin and against the grain, because a thick cold slice goes to rope between bread and a thin one stays tender and folds to the loaf. The meat has lost the warmth and the running juice it had on the plate, so the build has to put lubrication back: a measured smear of the horseradish, butter to the edges, or a thin layer of beef dripping that returns the roast's own fat 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 seal the crumb against the horseradish and partly to bridge the salt of the beef to the wheat. Cut thin and pressed, this is a sandwich whose quality is decided by three honest things in the right proportion, which is exactly why it works as the standard.
The variations swap the sharp counter or the bread and keep the cold-beef frame. English mustard replaces horseradish with a different heat; a creamed horseradish softens the edge for those who find the raw root too fierce. The same beef on a thick soft bloomer is its own reading, as is the Sunday-dinner version built with a Yorkshire pudding inside, and the hot salt-beef and dripping builds belong to their own traditions. Each deserves its own article rather than being crowded in here.