Yes, they eat them in Lombardy and elsewhere in northern Italy (as they do in France) in sausages (like the salame d'asino ragusano from Sicily) or as a stracotto.
It can also be a mild, creamy salty cheese, asìno, produced since the Middle Ages in the Friulian districts of Clauzetto and Vito d'Asio (hence its name), served on sliced of polenta, or melted with ham, or used to flavour pasta or fonduta.
Image by Jamain, Creative Commons License