The sausage and mash sandwich is a pub plate folded into bread the next day, and the mash is what defines it. The sausage stays the constant: pork bangers, fried or left over from the night before, set firm and split. The variable, the part that makes this its own sandwich rather than a sausage bap, is the mashed potato used as a soft bed inside the bread. Mash is not a side here and not a filler. It is doing structural work the bread cannot do alone: a layer of buttery mash laid against the crumb cushions the sausage, fills the gaps a round banger leaves, and turns a hard, rolling filling into something that sits flat and stable between two slices. It also carries the gravy. Cold sausages have lost the sauce that made them a dinner, and the mash, loosened with a little gravy, is how that sauce is put back inside a sandwich without flooding the bread.
The build works because the mash is engineered rather than simply spooned in. It is mashed smooth and kept stiff with butter rather than slackened with milk, because a wet mash slumps and soaks the bread while a firm, buttery one holds a shape and stays where it is put. It is spread to an even layer so the sausage presses into it rather than onto bare crumb, which spreads the load and stops the banger sliding out the back. The sausage is split lengthways and laid flat into the mash for the same reason it is in every version of this sandwich: a split banger sits stable and presents a browned face where a whole one rolls. The bread is soft and sturdy enough to carry a heavy, dense, slightly warm filling, buttered to the edges so the gravy in the mash does not go straight through, and closed with a gentle press so the two soft layers bind into one rather than shifting apart on the first bite. A thin stripe of extra gravy or brown sauce against the sausage is the acid-and-savour counter that keeps a soft, starchy, rich middle from reading as one flat note.
The variations branch off the same banger by changing what sits with it. Fried onions go into or onto the mash for a sweet layer; a fried egg adds a yolk; the same filling answers to a bap, a butty, a barm or a cob across the regional bread words. Each of those deserves its own article rather than being crowded in here.