Doria is a Japanese yoshoku dish of buttered rice under a creamy sauce, often a tomato or béchamel base, blanketed with cheese and baked until the top browns. The doria sando takes that gratin and rebuilds it between bread, which is the whole conceit and also the whole problem to solve. A rice gratin is wet, soft, and held together only by its sauce and its cheese; a sandwich asks it to behave like a filling. What defines this one is that the bread has to do the structural work the baking dish used to do, and the sauce has to be thick enough to be picked up rather than spooned. The bread and the doria need each other here in an unusually literal way: too thin a sauce and there is no sandwich, only a leak.
The craft is about controlling moisture without losing the dish. The rice is buttered and bound into a sauce stiff enough to hold a shape when cool, the cheese melted through it rather than only on top so the filling sets into a sliceable block instead of a pool. The bread is typically a sturdy roll or a thick shokupan, sometimes hollowed or toasted on the inside face so the crumb resists going soggy on contact. Some versions keep the doria warm and the bread crisp from a quick press; others set the filling firm and serve it cool like a sozai pan, a savory filled bread from a bakery case. A good one tastes of browned cheese and buttery rice with a bread that stays intact to the last bite. A poor one is a slumping wet mass, the crumb gone to paste and the rice sliding out the back, the failure that follows directly from skipping the thickening step.
The variations track the doria underneath. A tomato-sauced doria gives a brighter, sharper sandwich; a béchamel or seafood doria runs richer and softer and leans harder on the bread for contrast; a curry doria pushes it toward the kare end of the yoshoku shelf. Each of those underlying gratins makes a meaningfully different sandwich and deserves its own article rather than being crowded in here.