Put enough cold water to cover the fish in a pot with the fish, carrot, celery, and onion, and bring it to a boil over medium heat. As soon as the water boils, remove from the heat and let the pot stand for 5 minutes, then remove the fish with a skimmer and lay it on absorbent paper to drain and cool. When the baccala is cool enough to handle, remove all the bones, a necessarily manual operation. Do not remove the skin, however.

Put the milk in a pan over low heat, and heat it until it is hot, taking care not to boil or burn it. Meanwhile, crumble the baccala by hand and put it in a bowl. Use a pestle to pulverize the fish, stopping every now and then to whisk in, alternately, trickles of oil and hot milk. Go back and forth between using the pestle and whisk. Continue to beat tirelessly; the process should take about 15 minutes. Salt is not normally necessary, but do add some black pepper. If the mixture gets too cold to mix well during the mashing, reheat it in a bain-marie until it is sufficiently malleable. You should also reheat it until it's warm when you are about to serve it. Accompany it with croutons of fried white bread or fried polenta