The fried egg butty is the one where the runny yolk is allowed to be the whole event, and the word gives that away. Butty means buttered bread, a kitchen-table name with no ceremony in it, and calling a fried egg between soft bread a butty rather than a sandwich sets the expectation: the yolk is meant to break, run down the wrist, and be eaten over the sink or standing at the counter before it can be helped. Where the plain egg sandwich treats the yolk as something to be controlled, the butty treats the spill as the point. The egg is fried with the yolk soft and unbroken on purpose, and the soft bread and the butter exist to soak what escapes rather than to keep it in.
The craft is the yolk, the butter, and the timing, in that order. The egg is fried until the white is fully set but the yolk is still liquid, because a hard yolk in a butty is a contradiction in terms and a broken one denies the sandwich its only sauce. Soft white bread or a soft roll is chosen so it yields around the egg and accepts the yolk as it goes rather than fighting it, and the butter is structural here, spread to the edges as a thin fat layer that carries salt across the bland bread and gives the lean white something to ride on. Salt and pepper go straight onto the egg, generously, since the yolk is the only flavour the build has. It is closed, pressed gently, and eaten at once, because a fried egg butty does not survive being made and set aside: the yolk waits for nobody and the pleasure is in catching it warm and slack.
The variations are the everyday additions that ride along with the running yolk. Brown sauce or ketchup is the standard companion, its acid cutting the richness the way it does in any breakfast roll; a rasher of bacon turns it into the bacon-and-egg build with the yolk as the sauce for both; the plain sliced-bread reading, where yolk control rather than yolk spill is the aim, is its close and quieter relative. Each deserves its own article rather than being crowded in here.