The beef dripping sandwich makes a filling out of fat alone. Beef dripping is the fat rendered from roasting a joint, set firm as it cools and carrying the savoury, slightly meaty taste of what it came off. Spread thick on bread, sometimes with the dark jellied juices from the bottom of the pan, it is the whole sandwich: no slice of meat, just the rendered fat standing in for one. That is the defining logic and the reason it exists, a way of getting the flavour of a roast from the part of it that would otherwise be thrown away.
The craft is temperature and what comes with the fat. Dripping is hard from the fridge and greasy when warm, so it spreads cleanest at cool room temperature, where it goes on like a firm butter and stays put rather than soaking straight through or sliding off. The prized part is the brown jelly that settles under the white fat as the dripping sets, the concentrated meat juices, and a sandwich that reaches down to include that jelly carries far more roast flavour than the pale fat alone. Bread with real structure, a sturdy white or a bloomer, is chosen because a rich, dense spread wants a crumb that can hold it without collapsing, and a little salt over the dripping sharpens a filling that is otherwise all savoury depth and no edge. The bread is not toasted, because the point is the firm fat against soft bread, not a crisp base.
The variations are few and stay honest to the same rendered-fat idea. Salt and a grind of pepper is the standard finish. A scrape of the dark jelly worked through the fat is the richer build and the one most worth seeking. Cold beef scraps from the joint folded in push it toward a leftover-roast sandwich rather than a pure dripping one. Each deserves its own article rather than being crowded in here.