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Sprinkle sea salt and freshly ground pepper all over the loin. Pat it in with your hand.
Put the butter and oil in a enameled or heavy-bottomed pot that fits the loin snugly over medium-high heat.
When the butter foam subsides add the meat fat side down.
Brown the loin thoroughly on all sides. Lower the heat if the butter turns dark brown.
Slowly add the milk to the pot.
When the milk comes to a boil reduce the heat to medium-low or even low to keep the milk at a low simmer, cover the pot with the lid a bit askew.
Cook the loin slowly until the meat is fork-tender, about 1 ½ to 2 hours.
Turn and baste the loin occasionally and if needed add more milk.
By the time the loin is cooked the milk should have coagulated into small nut-brown clusters on the bottom of the pan. (If it is still pale remove the loin, uncover the pot, raise the heat and cook briskly until the milk bits darken.)
Remove the loin and let it rest for about 5 minutes before slicing.
Skim all the fat from the pot. Add a few tablespoons of water to the pan and scrape up all the residue on the bottom of the pot as the water evaporates. Taste the pan sauce and add more salt and black pepper if desired.
Cut the loin into half-inch slices and arrange them on a serving platter.
Spoon the pan sauce over the slices and serve immediately.