Preheat the oven to 350 degrees.
Liberally season the pork on both sides with salt and pepper.
Film a large Dutch oven with the oil and set over medium heat.
When it begins to shimmer, set the garlic cut-side down in the pot. Cook until golden, about 3 to 4 minutes. Remove the garlic and set aside.
Add the pork and cook until well browned and very crisp on all sides, about 15 minutes. Remove the pork and set aside.
Add the wine and stock to the pot and stir, scraping up all the browned bits on the bottom of the pot.
Return the pork and garlic to the pot and add the herbs. Cover and transfer to the oven.
Cook, turning occasionally and adding more stock if necessary, until the pork is very tender and falls apart when pulled with a fork, about 3 to 3 ½ hours.
Transfer the pork to a cutting board to rest for about 10 to 15 minutes.
Squeeze 2 of the garlic halves to pop the cloves into the braising liquid and discard their skins as well as the herb sprigs.
Remove the fat that forms on the top of the juices. Set the pot over medium heat and simmer until the juices reduce slightly. Season to taste.
Slice the pork against the grain into thick pieces and serve drizzled with braising liquid on top. Garnish with the remaining garlic halves.
