The panino con frico puts cheese in bread as a crisp, not a melt, and the ambiguity in the name is part of the sandwich. Frico is a Friulian preparation of grated Montasio, the firm Alpine cow's-milk cheese, and it exists in two forms: frico croccante, a thin disc fried until it sets into a brittle gold wafer that snaps, and frico morbido, a soft pan of cheese bound with potato and onion into something tender and dense. A panino con frico can be built on either, and which one decides the entire texture of the sandwich, since one filling shatters and the other yields.
The craft depends on which frico goes in. The crisp version is grated Montasio cooked in a hot pan until the fat renders and the proteins set, kept thin so it sets brittle through with no soft middle, and slid into the bread the instant it leaves the heat, because a cooled crisp picks up moisture from the air and the crumb and turns leathery within minutes. The soft version is the opposite: potato and onion cooked down with the cheese into a pliable cake that wants no hurry and behaves like a substantial filling, warm and rich rather than fragile. Montasio is the cheese in both because it has the fat and protein to render and reset cleanly; a half-cured wheel crisps without scorching and binds without going greasy. The bread is plain and soft in both readings, a neutral shell that carries either a fragile wafer or a heavy cake without competing with it.
The variations are the two frici and their close kin: the brittle croccante slid hot into a roll, the morbido bound with potato as a fuller filling, the version that scatters the crisp over a little cured meat. Each is its own preparation with its own texture, and each deserves its own article rather than being crowded in here.