Preheat the oven to 325°F. Trim away any large excess pieces of fat from the pork shoulder. If using a boneless cut, divide it into 4 large sections; if the pork is bone-in, it may be left whole. Pat the meat dry thoroughly with paper towels, then season all sides generously with kosher salt and freshly ground black pepper.

Place a large Dutch oven over medium-high heat and add the oil. When the oil is hot, arrange the pork in a single layer and sear until well browned on every side, about 4 to 5 minutes per side. Work in batches if necessary so the pot does not become crowded.

While the pork browns, combine the apple cider, chicken broth, Dijon mustard, and dried minced onion in a bowl and whisk until smooth. Tie the rosemary and thyme together with kitchen twine to create a small herb bundle.

Once the pork has developed a deep brown crust, pour the prepared braising liquid into the Dutch oven. Add the herb bundle and the halved head of garlic to the pot. Cover and transfer to the oven. Braise for about 3 hours, turning the pork halfway through cooking. For boneless pork, begin checking for tenderness at around 2 1/2 hours.

When the pork is nearly fork-tender, remove the pot from the oven. Tuck the thickly sliced red onion and the apple quarters around the meat. Cover again and return the pot to the oven for 30 to 45 minutes more, or until the pork is very tender and the fruit and onion have softened into the cooking liquid.

Remove the Dutch oven from the oven and allow the pork to rest in the braising juices for about 30 minutes. Press the softened garlic cloves from their skins into the liquid or over the pork. Taste the braising liquid and adjust with additional salt and black pepper as needed. Serve the pork with the apples, onions, and generous spoonfuls of the flavorful pan juices.