Square sausage and egg is the Lorne slab carrying a fried egg, and what defines it is how the two flat layers are made to work together. Lorne, or square sliced sausage, is a seasoned beef and pork mixture set in a loaf tin and sliced into flat slabs rather than piped into links, so it has a fine, close, even texture and a shape that covers the full face of a roll edge to edge. The egg is the second flat layer set against it, and the sandwich is built around managing two flat things, a firm savoury slab and a soft yolk, in one soft roll. The squareness is what makes that clean: a slice of Lorne lays flat under the egg in one even band where a round sausage would leave gaps and roll out of the bread.
The craft is in the griddle and the order of the two layers. The slab is cut to the width of the roll and cooked on a flat heat so both faces brown and the close-textured meat firms enough to hold its shape and not bend or slide when the roll is lifted. The egg is fried so the white is set firm enough to handle but the yolk stays liquid, because the runny yolk is doing real work here: it is the sauce and the lubrication, breaking over the dense sausage and binding the slab to the roll the way butter alone cannot. The roll is a soft floured morning roll, yielding enough to compress to the slab but sturdy enough to take the rendered fat and the running yolk without going to paste, and butter bridges the salt of the sausage across to the wheat. The one sauce, brown or red, goes inside in a measured stripe so it cuts the fat without burying either the seasoned meat or the yolk.
The variations are mostly about what is added to the slab and egg. A tattie scone underneath turns it into a fuller Scottish stack; black pudding adds a second savoury slab; bacon joins the band for a heavier roll. The plain square sausage roll drops the egg for the slab alone. The wider breakfast-roll shelf, the bacon roll under every regional name, the link-sausage bap, is the same morning idea on a different cut. Each deserves its own article rather than being crowded in here.