Heat the oven to 325 F.

Pat the pork roast with paper towels to dry. Trim and discard any visible fat.

Sprinkle the pork roast all over with salt and pepper. Set aside.

Heat 1 tablespoon of olive oil in a large heavy Dutch oven or saucepan over medium heat. Add the carrot, celery, and chopped onion. Cook, stirring frequently, until onion is tender. Add the garlic, rosemary, sage, and thyme. Cook for 2 minutes longer, stirring frequently. Remove the vegetables and herbs to a plate and set aside.

Add the remaining olive oil to the pot and turn the heat up to medium-high. Add the pork shoulder to the hot oil and cook for about 10 minutes, turning to brown on all sides.

Add the vegetables and herbs back to the pot with the pork shoulder. Add the apple cider and chicken broth; bring to a boil.

Cover the pot and transfer the pork to the oven. Braise in th oven for 3 hours, or until the pork is very tender.

Slice, shred, or chop the pork. Skim the fat from the juices, strain, and drizzle over the pork to moisten.