A piece is not a particular sandwich but the Scots word for the thing itself, and that is exactly what defines it. To ask for a piece, or to be sent out with one, is to ask for a sandwich, specifically one carried out of the house to be eaten elsewhere: at work, at school, in the back court, up the close. The filling is unspecified on purpose. A piece and ham, a piece and cheese, a piece on jam, a piece and sausage are all just a piece with the contents named after it. The identity here is linguistic and social rather than a recipe: the word marks a sandwich as portable, everyday, and Scottish before it says anything about what is inside it, and that is the whole of what makes a piece a piece.
The craft is the craft of food built to travel, read through the dialect that names it. Whatever the filling, the bread is buttered firm and to the edges because a piece is by definition eaten away from where it was made, and a slack, dry, or bleeding piece has failed at its one job long before anyone unwraps it. The filling is kept simple and the bread plain soft white, the loaf that was in the house, because a piece was never about refinement: it was a working sandwich, made from what there was, made to keep someone going between meals. The butter is structural as much as flavour, sealing the crumb against a moist filling so the piece survives a pocket or a tin until the break it was made for.
The variations are the larder and the regional word rather than a change of idea. A piece and ham, a piece and cheese, the jeely piece of jam and Glasgow tenement identity, the same carried sandwich met through whatever was in the press. The wider British map of the bap, the barm, the cob, and the butty is the same portable idea wearing a different region's word. Each deserves its own article rather than being crowded in here.