Egg and chive is the egg sandwich with its one failing addressed directly. Hard-cooked egg bound with mayonnaise is rich, mild, and faintly sulphurous, and that sulphur is the note that turns the filling heavy by the third bite. Chive is the answer. Snipped fine and folded through the bound egg, it lays a thin, clean allium line across the whole filling that cuts the sulphur and lifts the eggs out of their own richness without adding anything that competes with them. That is the entire reason the herb is in the sandwich and the entire reason it earns a name of its own: take the chive out and you have a plain egg mayonnaise, and the difference between the two is precisely the smell the chive is there to manage.
The craft is the bind first and the herb second. Egg is chopped and held with just enough mayonnaise to make it cohere, because too little and it falls out the sides and too much and it slumps into a slick, and that ratio is unforgiving since there is almost nothing else in the sandwich to draw attention away from it. The chive is cut fine and worked through the mixture rather than scattered on top, so every bite carries the same measured amount, and it is used in restraint: enough to read as a fresh green cut against the egg, not so much that it turns a gentle filling oniony. Salt and white pepper go through the egg in the bowl, not onto the bread. The bread is soft and plain and buttered to seal the crumb against a filling that is, by design, slightly wet, and the sandwich is pressed and cut so the herb-flecked egg holds in a clean face rather than spilling.
The variations are the rest of the things people fold into bound egg to break its softness, each a different counter to the same constant. Cress brings a peppery shoot instead of an allium; watercress brings the same pepper as a leaf with body; tomato brings acidity and the wet-bleed problem that comes with it. Each is its own pairing rather than a tweak to this one, and each deserves its own article rather than being crowded in here.