Heat the olive oil with 3 tablespoons of the butter in a large, heavy saute pan over a medium flame. Add the onions and cook them over low heat until they are very soft but not colored, for literally about 1 hour.
Salt the onions and remove them to a warm platter. Add the liver to the pan, salting and cooking for 30 to 45 seconds on each side. Work in batches if necessary to avoid overcrowding the pan. When done, place the liver over the onions and keep warm.
Add the wine to the pan and scrape the bottom of the pan with a wooden spoon to dislodge any browned bits. Add the remaining tablespoon of butter and pour it over the liver and onions. Drizzle the vinegar over and serve immediately.
Recipe copyright 2001, Mario Batali. All Rights Reserved.