The panino con Asiago is a single Veneto cheese given a bread, and which cheese it is depends entirely on age. Asiago DOP is really two cheeses under one name. Asiago pressato, the fresh form, is pale, soft, supple, and mild, a young table cheese that melts and yields easily. Asiago d'allevo, aged for months or years, is firm, dry, granular, and sharp, closer in register to a hard grating cheese than to the fresh one. The sandwich is a frame for one of those at its right ripeness, and the entire decision is which Asiago, because the two behave so differently that they make almost different sandwiches from the same denomination.
The craft is matching the form of the cheese to the bread and the dressing. The fresh pressato is sliced thick and laid in slabs, mild enough to be the quiet centre of the sandwich and soft enough that a little warmth from the bread starts it slackening; it wants a plain, fairly soft loaf so nothing argues with its gentleness. The aged d'allevo is the opposite problem: dry and assertive, it is cut thinner or shaved and met with something to answer its salt and bite, often a sturdier crusted bread and a touch of honey, fruit, or a few walnuts so the sharpness has a counter. The bread choice and the accompaniment are dictated by the age of the wheel, not decided in advance.
The variations stay in the cheese and the same fresh-versus-aged logic: the mild pressato build against the sharp d'allevo, the aged version finished with honey or mostarda, the one that pairs the cheese with the speck of the neighbouring mountains. Each of those is one expression of the wheel given a bread, and each deserves its own article rather than being crowded in here.