Sausage and egg is the banger sandwich with a fried egg added, and the egg is what the build is now about. The sausage stays the constant: pork bangers fried until the casing has burnished and the inside has set, split and laid in soft bread. The variable, the thing that changes the whole engineering, is the yolk. A runny yolk laid over a split sausage is not a second filling sitting politely beside the first. It is a sauce that has not been made yet, held in a thin membrane, waiting to break across the meat the moment the sandwich is pressed. That is the defining fact of the sandwich and the reason it behaves differently from a plain sausage bap or butty: there is now a liquid element inside that has to be timed, contained, and aimed, and getting that right is the entire craft.
The build is a yolk problem solved by how the egg is cooked and how the bread is chosen. The egg is fried for a fully set white and a still-liquid yolk, because a hard yolk leaves the sandwich dry and a loose white is structurally useless; the set white is what holds the yolk in place until it is wanted. The sausage is taken all the way through and split lengthways so it lies flat and stable, giving the soft egg a firm, browned, salty counter rather than softness meeting softness. Butter spread to the edges of the bread seals the crumb so the broken yolk and the rendered sausage fat do not soak straight through before the thing is eaten, which on a sandwich built in under a minute is a real constraint, not a refinement. The fold or the closed roll is pressed gently and evenly so the yolk breaks in a controlled spread into the bread and the meat rather than squirting out one side, which is the failure the whole assembly is arranged to avoid.
The variations are the rest of the breakfast packed around the same egg-and-sausage core. Bacon joins or replaces the sausage and adds a second rendered fat; fried onions bring a sweet counter alongside the yolk; the same filling answers to a bap, a butty, a barm or a cob depending on the region's bread word, and the brown-against-red sauce divide lands the moment the egg does. Each of those deserves its own article rather than being crowded in here.