The coleslaw sandwich is the rare British sandwich whose filling is also its sauce, and that is the whole problem to solve. Coleslaw is shredded raw cabbage and carrot bound in mayonnaise, so it arrives wet, cold, and crunchy with no separate dressing needed and no protein at all. Put between two slices on its own, it has to do three jobs at once: supply the bite, supply the moisture, and hold together long enough to be eaten in the hand. The defining fact is that there is nothing else in the sandwich to absorb or balance the slaw, so the slaw has to be made tight enough to behave as a filling rather than a salad that happens to be near bread.
The craft is moisture control above everything. Cabbage and carrot weep salt water and the mayonnaise slackens against bread, so a loose, freshly dressed slaw soaks the crumb to paste within minutes. A good coleslaw sandwich uses slaw that has been salted, drained, and bound only just enough to coat, so the vegetable crunch survives and the bind holds without running. The bread is soft and plain because the slaw already supplies the whole texture and flavour, and butter to the edges is structural here, waterproofing the crumb against the dressing so the sandwich does not collapse before it reaches the mouth. The cabbage is cut fine enough to bite cleanly rather than pull out in long strands, which is part of what keeps each bite the same instead of mostly bread at one end and a wet clump at the other.
The variations are mostly about what the slaw is asked to support. Cheese and coleslaw is the common pairing, the slaw acting as the moist, sharp counter to a dense block of Cheddar. Coleslaw with ham or chicken turns it into the cool dressing for a cold cut rather than the filling itself. A sharper vinegar-based slaw with no mayonnaise changes the register from creamy to bright. Each of those is its own sandwich and deserves its own article rather than being crowded in here.