The potato bread sandwich is a Northern Irish build where the bread is not a wheat loaf at all but fried potato bread, and that substitution is the whole sandwich. Potato bread, also called fadge, is the soda-farl-adjacent griddle flatbread of the Ulster fry, made from mashed potato, flour, and a little fat, cooked flat on a hot pan until it is soft inside with a faintly crisp surface. Here two slabs of it, usually fried in bacon fat, stand in for the slices of bread, with a fry-up filling, classically bacon and egg, held between them. The defining fact is that there is no leavened crumb anywhere in it: this is starch carrying starch, a fried potato base doing the job a bap or a slice would normally do, and the flavour follows from the carrier rather than the filling.
The craft is governed by what fried potato bread does and does not do. It has very little structural give and almost no absorbency next to a soft roll, so it does not soak up bacon fat the way a bap is engineered to; instead it carries that fat on its fried surface, which is why it is cooked in it in the first place, taking on salt and colour as it goes. The texture logic is reversed from a normal butty: rather than a crisp filling against a soft yielding crumb, this is the filling against a denser griddled potato slab, a firm carrier rather than a soft one. There is no butter step, because the frying fat has already done the bridging and seasoning work butter does in a roll. A fried egg with a runny yolk earns its place as more than filling here, the yolk supplying the lubrication the missing butter would otherwise provide and binding the firm slabs together. It is eaten hot off the pan, because there is no waterproofed crumb to hold it once it sits and cools.
The variations are the rest of the Ulster fry this is cut from and the regional griddle carriers that parallel it. Bacon and potato bread is the same construction named for its filling; soda farl is the other Northern Irish griddle base for the identical fry-up; the Scottish tattie scone is the closest relative across the water, a different region's fried potato carrier for the same idea. Each deserves its own article rather than being crowded in here.