A fried egg does something to a black pudding sandwich that no condiment can: it binds it. Fried black pudding has a crumbly, slightly sandy texture, the oatmeal and set blood breaking apart rather than holding together, and a sandwich built on it alone tends to shed its filling at the first bite. A soft yolk broken over the hot slices runs into that crumble and glues it, turning a loose, fragile filling into a single coherent layer. The egg is the variable that solves a structural problem the pudding creates, and it does it with richness rather than with the acid an apple would bring.
The craft is the state of the yolk and the order of assembly. The pudding is sliced and fried hard so its faces crisp and the centre stays soft; the egg is fried so the white is set but the yolk is still liquid, because a hard yolk binds nothing and a raw one floods the bread. The egg goes on top of the hot pudding so the yolk breaks down into the crumble rather than pooling against the crust, and the sandwich is closed and eaten at once, while the yolk is still doing its work. The bread is soft and plain, a floured roll or buttered white, sturdy enough to take a wet, rich filling without going to paste. Butter underneath seals the crumb against the yolk and carries the pudding's spiced salt across the slice.
This is the binding answer to what black pudding wants beside it; the others answer differently. Apple cuts the iron with sweet acid instead of softening it with fat; bacon adds salt rather than richness; brown sauce and a flat mushroom carry the whole thing toward the full breakfast. Those deserve their own articles rather than being crowded in here.