The toast al salmone takes the format of the Italian griddled bar sandwich and swaps its usual cooked ham for smoked salmon, which changes what the heat is allowed to do. The defining fact is that the salmon is the one element that must not really cook. Salmone affumicato, cold-smoked and silky, turns dull and rubbery the moment it sets, so this build asks the press to brown the bread and soften the cheese while leaving the fish barely warmed through. The cheese is what carries the structure here: a soft fresh spread or a mild melter that binds the slim parcel together and gives the lean, smoky fish a fat to lean against. Without the cheese the salmon has nothing to round its salt and smoke; without restraint at the iron the salmon is ruined. The pairing works only when the bread does the cooking and the salmon mostly does not.
The craft is heat control and the order of assembly, because the margin between warmed and overcooked salmon is thin. The bread is soft pane in cassetta, buttered on the outer faces so it crisps gold in the clamp, and the cheese, often a stracchino-style fresh cheese, a light cream cheese, or a young soft melter, is spread on the inner faces so it forms a seal and a buffer. The salmon goes in cool and in a single even layer, folded rather than piled, so it heats gently rather than frying. Many good builds barely press at all, just long enough to set the crust and warm the seam, and a thread of lemon or a few capers or a turn of dill is laid in to cut the richness. A good toast al salmone comes off the heat with a crisp shell and salmon that is still soft and translucent at its centre; a sloppy one is clamped hard and long until the fish goes opaque and tough and the smoke turns acrid.
The variations are one-element swaps against the same pressed white base, and each deserves its own article rather than being crowded in here. There is the classic prosciutto cotto and cheese build that the format is named for, the four-cheese version with no fish or meat at all, the relative that uses smoked trout in place of salmon, and the cold un-pressed version closer to a tramezzino than a griddled toast. Each is the same logic with the protein or the heat changed.