The brawn sandwich is held together by something that is already set before it reaches the bread. Brawn is meat from a pig's head, slow-cooked until it falls from the bone, then pressed into the gelatine-rich stock it released and chilled until the whole thing sets into a sliceable terrine. That jelly is not a garnish around the meat; it is the matrix the meat is suspended in, and it is what lets a collection of soft, shredded offcuts behave like a firm cold cut you can lay between two slices of bread. The defining fact is that the structure comes from the filling, not the sandwich: brawn arrives pre-set, and the bread is only carrying a slice of something that already holds its own shape.
The craft is keeping that set intact and cutting through the richness. Brawn is fatty, gelatinous, and faintly sweet from the long cook, so it is sliced cold and kept cold, because warmth slackens the jelly and a softened slice slumps out the sides rather than sitting square. The bread is soft and plain so it yields to a delicate, wobbling filling instead of compressing it, and butter on the bread bridges the mild meat to the wheat and seals the crumb. The non-negotiable partner is acid: a sharp mustard, a vinegary pickle, or piccalilli, applied as a stripe rather than a flood, because brawn is all soft savoury depth with no edge of its own, and without something sour to push against it the sandwich reads as one rich, jellied note. The bread is not toasted; the contrast that matters is cool, set meat against soft crumb, not crispness.
The variations stay inside the cold, set, sharply countered frame. Tongue and other pressed and potted cold cuts share the same logic of a meat that sets firm enough to slice. A more heavily seasoned brawn, peppered or studded with herbs, changes the cure but not the structure. Headcheese under its other regional names is the same terrine wearing a different word. Each deserves its own article rather than being crowded in here.