The corned beef hash sandwich is the only one of the corned beef family that is hot, and that changes everything about it. Where the plain sandwich lays cold tinned beef between slices, hash takes that same salt-cured beef, breaks it up, and fries it hard with diced potato until the potato catches a crust and binds the loose meat into a single craggy mass. The potato is not a side here; it is the binder and the bulk, soaking up the rendered beef fat and turning two cheap things into one substantial filling. The defining fact is that this sandwich starts as a leftover dish in its own right and is then put into bread, so what goes between the slices is already cooked, already bound, and still warm.
The craft is the fry and the containment. The hash has to be pressed and left alone in a hot pan long enough to take colour, because the crisp brown crust is the whole textural argument against soft bread; a pale, stirred hash is just a damp savoury paste. The mixture is also hot and greasy, so the bread is plain and soft and buttered to the edges to take the fat without dissolving, and it presses down onto the hash to hold the loose fried mass together for the few bites it lasts. Unlike its cold relatives this sandwich wants no sharp pickle worked through it; the contrast that matters is already built in, crisp potato edge against soft fried beef, and a fried egg laid on top, its yolk run into the hash, is the more common addition than any acid.
The variations stay inside the hot, potato-bound frame. A heavily peppered or chilli-spiked hash changes the seasoning without changing the structure. A version run with leftover roast or boiled potato eats softer than one built from raw diced potato fried from cold. The fried egg on top is common enough to be almost standard but is its own decision. The cold corned beef sandwich, and its onion and pickle versions, are a different family entirely and deserve their own articles rather than being crowded in here.