The hog's pudding sandwich is a West Country sausage problem, and the variable that defines it is a spiced pork-and-grain pudding that has to be cooked before it can go anywhere near bread. Hog's pudding is a Cornish and Devon sausage of pork, fat, suet, and groats or oatmeal, seasoned heavily with pepper and mace and other warm spice, sold in a length and never eaten raw. It is sliced into thick discs and fried until the outside crisps and the grain-bound interior heats through and softens, then laid hot into a roll. That is the founding fact: unlike the cold cuts on the same counter, this sandwich does not exist until heat has been applied, and its whole character is the contrast between a fried, peppery, slightly coarse pudding and a soft, plain bread.
The craft is the fry and the bread that receives it. The pudding is cut thick enough that the disc holds together in the pan and keeps a soft, mealy centre under a crisped face, because sliced thin it dries out and loses the textural split that makes it worth doing. It goes into the roll hot, while the fried edge is still firm, so a soft bap or a thick slice of plain bread is chosen to take some of the released fat and yield around the discs without competing with the spice that is already strong inside the meat. Butter or the pan fat is the bridge across the crumb. Mace and pepper are doing the seasoning from within, so the sandwich wants little on top: at most a sharp sauce or a smear of mustard as a counter to the richness, applied as a stripe rather than a wash, since the pudding is loud enough on its own.
The variations stay inside the fried, spiced, West Country frame. A brown sauce or English mustard alongside is the usual sharpening rather than a different sandwich; a fried egg over the discs turns it into a fuller plate folded into bread. The wider family of cooked grain-and-pork puddings, white pudding and the oat-bound Scottish and Irish kin, shares the slice-and-fry logic with a different spicing. Each deserves its own article rather than being crowded in here.