The muffuletta is held together by a salad, not a sauce. Salami, mortadella, ham, and provolone are layered on a wide, round, sesame-crusted loaf, and the component that defines the sandwich is the olive salad: a chopped, marinated mix of green and black olives, giardiniera vegetables, garlic, and herbs in oil that is spread thickly across the bread on both faces. That oil is the entire point. It soaks down into the round loaf and seasons it from the inside, so the bread stops being a neutral carrier and becomes part of the filling. A muffuletta without the olive salad is just cold cuts on a roll; the salad and its oil are what give the sandwich its name and its logic.
It works because the round seeded loaf is built to absorb that oil and survive it. The bread is sturdy and slightly chewy with a crumb dense enough to take a heavy, oily dressing without collapsing into paste, which a soft sub roll could not do. The build order matters: the olive salad goes against the bread on both inner faces, not buried in the middle, so the oil has a direct path into the crumb from top and bottom rather than being trapped between layers of meat. The sandwich is assembled and then deliberately left to sit, often wrapped and weighted lightly, so the oil migrates evenly through the loaf and the layers settle into one another, which is why it improves with a rest rather than demanding to be eaten the second it is built. The cured meats are shingled thin so each bite gets salami, mortadella, and ham together rather than a slab of one; the provolone binds and softens against the warm-from-resting bread; and the marinated vegetables cut the doubled richness with acid and crunch from inside the structure rather than as a topping. It is cut into wedges from the round, and each wedge is sized to hold all of it, meat, cheese, and oil-soaked bread, in a single bite. The press is optional and regional; the salad is not.
The variations stay close to the round-loaf, olive-salad frame and each deserves its own article rather than being crowded in here. Some kitchens serve it warm and lightly pressed so the provolone softens into the meats; some lean the olive salad more heavily toward giardiniera or more heavily toward olives, which shifts the whole sandwich's acidity. The muffuletta sits within the dense long tail of regional American specialties, sandwiches tied so tightly to one city that they barely exist a state away, and its New Orleans identity is inseparable from the build. Each of those regional readings is one adjustment on a fixed idea, which is the impulse that earned the muffuletta its own name.