Preheat oven to 300°F. Pat the pork dry and lightly score the fat cap.
Stir salt, pepper, brown sugar, thyme, rosemary, sage, and garlic into a paste. Rub it all over the pork.
Heat a large Dutch oven over medium-high. Add olive oil and brown the pork on all sides, 6–8 minutes. Transfer to a plate.
Add onion to the pot and cook, stirring, 3 minutes. Stir in apples and cook 2 minutes more.
Pour in apple cider and simmer 2 minutes, scraping up browned bits. Stir in broth, vinegar, and Dijon.
Nestle pork fat-side up on the onion-apple mixture. Cover tightly and roast until fork-tender, about 4 hours 15 minutes.
Transfer pork to a board and rest 15 minutes. Skim fat from the pot juices and strain if desired.
Melt butter in a saucepan over medium heat. Whisk in flour and cook 1 minute. Slowly whisk in hot cooking liquid and simmer until silky, 5–7 minutes. Season to taste.
Slice or shred the pork. Serve with warm apple gravy over the top.