Ham and egg leads on the egg, and specifically on what kind of egg, because that single choice is the whole sandwich. Cooked ham gives the build its salt and its structure; the egg is the soft addition that decides everything else. Sliced hard-boiled egg keeps the sandwich cold and firm, a lunchbox object that travels: pale, mild, slightly chalky discs laid over the meat. A fried egg makes it a different thing entirely, hot, with a yolk that has to be managed, the sandwich eaten now rather than carried. Same name, two sandwiches, and the soft component is the variable the cook is actually choosing between.
The craft depends on which egg won. With sliced boiled egg the problem is blandness and slip: the egg is mild and the rounds shed off the ham, so it wants salt and pepper worked over it and a measured smear of butter or mayonnaise to glue the layers and stop the discs sliding out the side. With a fried egg the problem is the yolk. A hard-set yolk makes the sandwich dry and defeats the point; a running yolk makes it delicious and structurally unstable, soaking the bread the moment it is bitten. The usual answer is a yolk set soft but not liquid, the egg built straight onto the ham while hot, and a soft roll sturdy enough to take the bleed. Either way the ham is the salt and the firm anchor under a deliberately soft top.
The variations stack the breakfast. Ham, egg, and chips folds the chip butty into it; ham and egg with brown sauce adds the morning condiment; bacon and egg swaps the cured meat and changes the salt; an egg mayonnaise base trades the sliced egg for a bound one and goes back toward the lunchbox. Each of those is its own sandwich and deserves its own article rather than being crowded in here.