The egg banjo is defined by its failure mode, which is also its name. A fried egg with a soft yolk is put in a roll and bitten, the yolk bursts and runs down the hand and the forearm, and the eater's reflex is to hold the roll out and wipe the drip down their sleeve with the free hand, an action that looks exactly like strumming a banjo. The British military gave it that name, and the name is not decoration: it is an honest description of what this sandwich does. Every other egg sandwich on the shelf is engineered to contain the egg. The banjo is the one that does not even try, and its whole character is that the runny yolk is the point, not a defect to be managed away.
The craft, such as there is, is minimal on purpose and entirely about the yolk. There is no bind, no mayonnaise, no salad, often nothing but a fried egg and a roll, sometimes a stripe of brown or red sauce. The egg is fried so the white sets but the yolk stays liquid, because a set yolk would make a tidier sandwich and a worse banjo. The roll is soft and absorbent so it takes some of the yolk rather than shedding all of it, and it is held and eaten in a way that accepts the mess as part of the deal. This is field food and canteen food, made fast with whatever is on the stove, and its design philosophy is the opposite of the meal-deal triangle: not survivable, not portable, just hot and immediate.
The variations stay inside the fried-egg-in-a-roll idea and mostly add to it. A rasher of bacon or a sausage turns it toward the full breakfast roll; brown sauce or red sauce is the standing national split; egg and sausage is the same register with the egg no longer alone. Each deserves its own article rather than being crowded in here.