Lorne sausage is a British sausage with a very different form factor to the other meat sausages.
Most sausages in Britain are of the link sausage type, where (traditionally) an intestine was filled with the sausage meat, providing a tube of 1cm - 2cm in diameter which was twisted to create a knotty stoppage in the intestine every 4" to 5".
In England, traditionally sausages == pork link sausages.
There has been a trend within the last decade or maybe two for gourmet link sausages, of Venison and Cranberry, or jugged hare and goose liver pate, but these seldom are found on the plate of a full cooked breakfast, even when the breakfast is served as a brunch.
In Scotland, there have also always been beef link sausages and, even better, the beef Lorne sausage.
In the beef Lorne sausage, aka 'flat sausage' or 'sliced sausage', the sausage meat is extruded in a shape approx. square, and approx. 3" to 4" along each side. There is no wrapper, whether of the traditional gut or of a modern substitute.
Like most components of the full cooked breakfast, the Lorne sausage can be grilled - but has the greatest artery-clogging effect when shallow fried.
In Scotland, where bread takes many forms other than the UK-wide white pan loaf, flat slices of beef Lorne sausage are to be found in round bread 'rolls'.
In the USA, some brothers of Scottish origin started serving a highly standardised mutation of the roll and sausage in their eponymous diner, 'McDonalds', which they grew into a local chain of diners before selling the right to franchise the diner in the rest of the USA (to some guy whose name eludes me - but you can find out who in Bill Bryson's book 'Made in America').