Everything here follows from the state the cod is in by the time it reaches the bread. Morue is cod that has been split, salted, and dried hard as a board, then soaked for a day or more in changes of cold water until it softens, swells, and turns mild again. By the time it goes into a sandwich it has been poached or flaked, sometimes pounded with olive oil and garlic into the Provençal brandade, sometimes simply broken into pale lobes. The bread is a sturdy split loaf with a real crust, and the filling is the rehydrated fish, the olive oil it was worked with, and not much that competes with it.
The logic is about salt and moisture. Cured cod, even after a long soak, keeps a faint brine and a clean sea flavor that sits well against plain wheat, so the sandwich does not need a sharp cheese or a heavy sauce to feel finished. What it needs is fat to carry the fish, which is why olive oil does most of the binding work, soaking into the crumb and softening the line between bread and filling. Flaked cod brings no structure of its own, so the crust has to hold the shape; the fish has to be moist enough that it does not go dry and stringy against the bread, which is the failure mode of cod that was under-soaked or over-poached. A few drops of lemon and a turn of black pepper are usually as far as the seasoning goes. Eaten in the hand, slightly cool rather than cold, it reads as a savory, oil-rich sandwich with a saline backbone.
Variations stay close to the cod itself. The brandade version spreads the emulsified cod, garlic, and oil thick like a paste rather than laying flakes; a version with a layer of sliced potato bulks the fish out and softens its salt; a few add a stripe of garlic mayonnaise or a leaf of frisée for an acidic counterweight. Each is a way of carrying the same cured fish on the same bread. The Sandwich à la Morue belongs with the fish sandwiches the catalog groups under Baguette Poisson, the tradition that runs from smoked river fish to tinned tuna. Its specific contribution is a fish that arrives dried and salted and has to be coaxed back to softness before it can be a sandwich at all.