Haggis and egg is the haggis roll solved for its one structural weakness, and the solution is a yolk. Haggis on its own in a roll is a warm, peppery, oat-bound crumble: fully cooked, fully seasoned, and loose enough that it tends to break apart and fall out the back of the bread. A fried egg laid on top of it is not there for richness alone. The runny yolk is a binder. When it breaks it runs down through the crumble and sets it into something that holds together for a bite, gluing the loose oat-and-offal mass to itself and to the roll. This is the Scottish breakfast reading of haggis, and the egg earns its place by doing a job the haggis cannot do for itself.
The craft is timing the yolk and managing two soft elements. The egg is fried so the white is set firm enough to lift but the yolk stays liquid, because a hard yolk binds nothing and the entire point is the flow. The haggis goes in hot and loose so the warm yolk can work through the crumble rather than sitting on a cold cap of it, and the two are assembled and eaten quickly, before the yolk firms and the haggis tightens. The roll is a soft floured morning roll, plain on purpose: it has to absorb a little fat and yolk without collapsing, and an assertive bread would fight a filling that is already peppery and now also rich. The haggis brings its own seasoning, so this is rarely sauced; the egg is the only addition, and it is addition enough, because it changes the sandwich from a crumble you chase to a bound, cohesive thing.
Haggis and egg is the breakfast member of the small Scottish haggis family, the one whose variable is a binding yolk. Its near relatives are set by what else joins the egg: square or Lorne sausage alongside it pushes it to a full fry, a tattie scone underneath adds a griddled base, brown sauce over the top adds a sharp wet note. The plain haggis roll and the neeps and tatties roll are separate builds. Those deserve their own articles rather than being crowded in here.