The Derbyshire oatcake is not a biscuit but a soft savoury pancake, and that is the thing to get straight about it before anything else. It is a thin, flexible round of oatmeal and flour batter cooked on a griddle, closer to a flannel than to a hard oatcake, and it is used the way a flatbread is used: filled, folded, and eaten in the hand. It sits very near the Staffordshire oatcake, with which it shares the same family and the same idea, the Derbyshire version run from a slightly different batter so it reads a touch firmer or lighter depending on the maker. The point is that the bread here is soft and foldable rather than sliced and stacked, and the whole sandwich is built on that pliancy.
The craft is in the batter and the fold. The oatmeal makes the round nutty and slightly coarse, and it has to be cooked through but kept soft, because an oatcake taken too far on the griddle dries and cracks the moment it is folded round a filling. Warm, it bends without splitting and steams the filling gently; cold, it stiffens, which is why the classic move is to heat it again before it is loaded. The filling is the savoury, slightly greasy kind a foldable bread can hold, cheese melted into the round, bacon, sausage, egg, and the round is folded over rather than topped so it contains the load one-handed. There is no butter as a rule, because the oatmeal carries its own body and the filling brings the fat.
The variations are mostly what goes in the fold. Grated cheese melted through, the most common; cheese with bacon or sausage for a full breakfast in a single round; a sweet reading with butter and syrup that turns it toward a pudding. The Staffordshire oatcake is its close cousin from the next county over, the same idea on a marginally different batter, and each of these deserves its own article rather than being crowded in here.