The carrot sandwich runs on thrift logic: it is the answer to the question of how to make a filling out of the cheapest, most available root in the kitchen when there is nothing richer to reach for. Grated carrot is bound with a little mayonnaise and spread on buttered bread, and that is the whole sandwich. There is no protein and no expense, which is exactly the point of it. It is the rationing-era and make-do reading of a sandwich, built to turn a vegetable everyone could grow into a complete lunch by giving it just enough fat and bread to stop being only a vegetable.
The craft is entirely a moisture and sweetness problem, because raw grated carrot brings two awkward properties to bread: it is wet and it is earthy-sweet with no salt or acid of its own. Grated coarse it sheds water as it sits and soaks the crumb to a soggy mat within the hour, so it is grated and then pressed dry, bound only lightly so the mayonnaise coats rather than slackens it, and sealed off the bread with butter to the edges that waterproofs the crumb and supplies the salt the carrot lacks. The sweetness has to be cut or it reads as flat and one-note, which is why a squeeze of lemon, a few sultanas, or a fold of cress so often comes in: not as decoration but as the acid, the contrast, or the crunch that an all-soft, all-sweet filling needs to become a sandwich rather than damp orange shreds.
The variations stay inside that grated-vegetable, make-do frame. Carrot with raisin or sultana leans the sweetness into a deliberate sweet-savoury balance; carrot with cress or chopped peanut adds the textural counter the plain version goes without; carrot folded into a wider salad sandwich with beetroot or cucumber spreads the moisture problem across more components. Each of those is its own balance to strike and deserves its own article rather than being crowded in here.