A piece and cheese is the Scottish word for a sandwich doing the plainest job a sandwich can do. In broad Scots a "piece" is a sandwich, the thing carried in a pocket or a poke to school or to work, and a piece and cheese is that thing at its most reduced: bread, butter, and a slab of cheese, named in the order it is built and the dialect it is built in. The naming is half the point. This is not a cheese sandwich dressed up with a regional label; it is a plain piece whose entire identity is the single filling it carries and the word the country uses for it.
The craft is in the cheese and the restraint around it. The cheese is cut thick rather than grated, usually a firm Scottish Cheddar or a local Dunlop, because the filling has to have presence when there is nothing else in the sandwich to lend it any. Butter goes edge to edge and does structural work as much as flavour work: it bridges the salt of the cheese to the plain wheat of the bread and seals the crumb so a cool, slightly oily slab does not slide. The bread is soft plain loaf, white or brown, because a piece is a carrier and a crust with real chew would only argue with a filling that has no job but to be cheese. Cut in half, wrapped, and pressed flat in a poke, it is engineered for a lunchbox and a long morning, not for a plate.
The variations are small and stay honest to the single-filling idea. A scrape of pickle or a smear of chutney turns it into a piece and cheese and pickle, the most common addition and the one that tips it toward a fuller sandwich. Brown sauce is the savoury alternative to pickle and a genuine kitchen-table preference. Its closest relative is the piece and ham, the same plain piece built around a slice of cooked ham instead of a slab of cheese, and the wider British cheese shelf runs to dozens of named cheeses handled the same way. Those deserve their own article rather than being crowded in here.