The toast prosciutto e formaggio is the Italian bar sandwich at its most settled: two slices of soft white pane in cassetta, a layer of prosciutto cotto, a layer of cheese, pressed flat and griddled until the outside is gold and the inside has gone molten. The defining fact is that this is a melt, not a stack. The cheese is not there to sit between the ham and the bread as a third flavour; it is there to liquefy under the press and glue the whole thing into a single sealed object that can be cut on the diagonal and held without falling apart. Prosciutto cotto, the gently spiced cooked ham, brings a clean mild salt and a tender bite that the heat barely changes. Take the press away and it is a cold ham sandwich; use a cheese that will not flow and the seam never closes. The two halves are built to fuse.
The craft is the press and the cheese choice, because everything depends on the interior melting before the crust burns. The bread is pliable sandwich bread with the crusts often left on, buttered on the outside faces so the contact surface browns evenly in the hot plate clamp rather than scorching in patches. The cheese is a good melter, usually a mild fontina-style or a soft sottiletta-grade slice or a young Emmental, laid in enough quantity to bind but not so much that it floods out the sides and carbonises on the plate. The ham goes in a single folded layer, not stacked thick, so the parcel stays slim enough to heat through. A good toast comes off the iron with a crust that cracks slightly and an inside that pulls into a thread when split; a sloppy one is pressed cold so the centre stays cool while the bread blackens, or overstuffed so it splits its seam and weeps fat onto the griddle.
The variations are mostly swaps of one of the two fillings against the same pressed-white-bread base, and each deserves its own article rather than being crowded in here. There is the smoked-salmon build that trades the cooked ham for cured fish and a soft cheese, the four-cheese version that drops the meat entirely and lets a blend of melters carry the whole thing, the crudo relative that uses raw cured ham added after the press so it is not cooked, and the open toast finished under a grill rather than clamped. Each is the same melt logic with one element changed.