The pie sandwich is the generic Northern habit of eating a meat and potato pie inside bread, and what defines it is that it is the unbranded version of the idea the pie barm made famous. Where the pie barm is specifically a Wigan pie in a barm cake, the pie sandwich is the same reasoning without the local name attached: a whole pie, shortcrust top and bottom, split into a buttered roll or pressed between two slices, eaten across the North as an everyday thing rather than a regional signature. The defining fact is the same stack of starches, pie lid, pie base, bread, but here it reads as ordinary working food rather than a town's calling card, which is the whole difference between this and its more celebrated cousin.
The craft is the carbohydrate-on-carbohydrate logic that also runs the chip butty and the pie barm. The bread is soft and plain because the pie supplies the structure, the seasoning, and the gravy-thick filling, and any real crust would only add another layer of resistance to something already dense. Butter is structural: it lubricates a dry pastry shell that carries no sauce of its own and stops the bread becoming a third dry crust against the pie's two. The pie goes in warm so it softens enough to bite through cleanly, because cold pastry inside cold bread is two arid things with nothing to bind them. Sliced bread presses flatter against a pie and holds it more evenly than a domed roll, which is why the plainest version of this is often a pie folded into a buttered slice rather than a roll at all.
The variations are the pie and the bread. Meat and potato is the default; a steak or a cheese and onion pie swaps the filling; a barm, a bap, or a plain slice changes the carrier and, with it, the regional word. The wider pie family, the pie barm, the pasty bap, the bridie, shares the same engineered-container logic. Each deserves its own article rather than being crowded in here.