The Cumberland sausage sandwich is shaped by the fact that the sausage does not come in links. A Cumberland is sold and cooked as one long continuous coil, coarsely chopped rather than minced, and seasoned heavily with pepper and herbs. That coil is the design decision the sandwich is built around: instead of fitting round links into a roll, the cook cuts a length straight off the spiral and lays it flat, so the bread carries an even band of meat the full width of the slice rather than gaps between sausages. The coarse cut and the pepper are the other half of the identity. This is not a smooth breakfast sausage; it is a rough, savoury, distinctly peppery one, and the sandwich exists to carry that seasoning rather than soften it.
The craft is in the cut and the fat. A length sliced off the coil is split or flattened so it sits low and does not roll out of the bread, and it is browned hard so the coarse meat firms up and the herb-and-pepper crust develops. The roll or bread is soft enough to yield to a substantial filling but sturdy enough to take the rendered fat without going to paste, and butter bridges the salt of the sausage to the bread while sealing the crumb against the grease. The one condiment is a sharp counter, brown sauce or a sharp mustard, applied inside in a measured stripe so it cuts the richness without flooding the bread or burying the pepper the sausage is prized for.
The variations are mostly about what is set against the coil. A fried egg whose yolk has to be managed; caramelised onion for a sweet counter to the pepper; apple sauce borrowed from the pork tradition. The other regional sausage sandwiches, the Lincolnshire with its sage, the Glamorgan that is cheese and leek behaving like meat, the Lorne square sliced flat, are each a different seasoning on the same idea and deserve their own articles rather than being crowded in here.