Preheat the oven to 325°F.

Season the pork generously all over with salt and pepper. In a Dutch oven or other large, ovenproof pot over medium-high heat, melt 1 tablespoon of the butter with the oil. Add the pork, fat side down, and sear for 3-4 minutes. Repeat to brown all sides, about 15 minutes total. Transfer the pork to a platter and pour off the fat from the pan.

Return the pot to medium heat and melt the other 2 tablespoons butter. Add the chopped sage leaves and cook for 1 minute. Stirring constantly, slowly pour in the milk and cream. Return the pork to the pot, fat side up, and add the bay leaves, lemon zest and 1 teaspoon salt. Season generously with pepper. When the liquid begins to simmer, partially cover the pot and transfer to the oven to cook for 2 hours, turning every 30 minutes (I didn't do this, just FYI).

Uncover and continue to cook until very tender, 30-60 more, again turning after 30 minutes if cooking for longer than 30 minutes more. Transfer the pork to a platter and let the pan sauce settle.

With a slotted spoon, remove the curds from the sauce, leaving behind the fat, and transfer to a fine-mesh sieve set over a small saucepan. Using a wooden spoon, push the curds through the sieve to make a smooth, creamy sauce. Warm the sauce over low heat and stir in the lemon juice. Taste and adjust the seasoning. The sauce should be subtly tart from the lemon and slightly peppery, to cut the pork’s richness. Cut the pork into thick chunk and serve at once, passing the sauce at the table.