Preheat the oven to 275 F. Season the pork loin with salt and pepper.
Heat the oil in a large pot with a lid over medium-high heat. Add the pork and brown for about 10 minutes total, turning it every two minutes or so as the underside browns. Transfer it to a plate. Pour off any excess fat from the pan.
Return the pan to medium-high heat and add the butter. When the butter is melted, add the leeks. Sauté for 5 minutes until the leeks start to soften. Add the crushed garlic cloves, and sauté for 2 minutes until the cloves turn lightly golden. Pour in the warm milk and add the lemon zest, sage, thyme, and bay leaves. Bring to a simmer, and tuck the pork back into the sauce fat side up (the sauce should come about ⅔ the sides of the pork). Cover and transfer the pot to the oven, turning the pork every 45 minutes or so, for 2 ½ hours. Remove the lid and continue to cook, uncovered for about 30 minutes or so until the pork and leeks are very tender. The internal temperature of the pork should read 140 F on an instant-read thermometer.
Transfer the pork to a cutting board and let rest for 10 minutes before slicing. While the pork is resting, remove and discard the zest and herbs, and continue to simmer the sauce in the pot over medium-high heat until it becomes a thick sauce. The sauce may look slightly broken or curdled; this is fine. That's what happens when dairy is cooked for a long while. If you prefer (and that’s what I did), you can pour the sauce into a blender or food processor and puree until smooth. If the sauce looks appealing as is, you can skip that step and just leave the vegetables intact in the sauce.
Thinly slice the pork, and serve warm with the sauce.
Notes
To crush garlic, simply place it on a cutting board and place the side of a large heavy chef’s knife on a couple of cloves at a time. Use the bottom of your fist to smack the side of the knife so that the blade crushes the garlic beneath. Repeat with the rest of the cloves.
The milk slowly caramelizes during the braising process, enhancing the natural sugars in the milk. The natural lactic acid helps to break down the pork, making it enticingly tender. The sauce will likely start to break apart or curdle — it's supposed to!
Puree the sauce if you want it to be creamier. Leave it as is if you prefer bits of caramelized sauce and sauteed leeks throughout.