1. Make Sachet d'Epices: Wrap the thyme, parsley, and peppercorns in a small piece of cheesecloth and tie with twine to form a sachet.

2. Brown Pork: Heat the oven to 400 degrees. Use paper towels to pat pork dry, then season generously with salt and pepper. Heat a large Dutch oven or other pot with a tight-fitting lid over high heat for 2 minutes, then add enough oil to barely coat bottom of pot and heat until shimmering. Cook the pork until well browned on all sides, turning with tongs once each side is seared (remember not to move the meat too soon or it will not brown properly and will stick to the pot; wait until it releases easily). This will take a total of 12 to 15 minutes; reduce the heat if the bottom of the pot is getting too dark (you want browned, not burned, bits for flavoring the sauce). If, after removing the pork, you see burned bits, wipe out the pot and add more oil before proceeding. (Or deglaze pot with a little water, bringing it to a boil and scraping up the burned bits; pour off liquid and bits.)

3. Cook Aromatics: Reduce heat to medium and add leek, garlic, parsnip, and celery root. Season with salt and pepper. Stir frequently and cook until leek is translucent, about 2 minutes.

4. Braise Pork: Return pork to pot, and pour in 1 cup cider. Bring to a boil, and deglaze pot, scraping up browned bits from bottom. Add remaining 3 cups cider and the stock along with the herb sachet. (The liquid should come about halfway up the sides of the pork; add more stock if it doesn't.) Bring to a boil on top of the stove. Cover, and put in the oven. Reduce oven temperature to 325 degrees. Cook until the pork is very tender (it should offer little resistance when pierced with a knife), 2 to 2 1/2 hours, turning over with tongs about halfway through so the meat cooks evenly.

5. Finish Braising with Garnish Vegetables: Transfer the meat to a plate and strain the braising liquid through a fine sieve, pressing on the solids to extract as much liquid as possible (discard solids). Return the liquid and the pork to the pot and add the garnish vegetables, nestling them into the liquid; the liquid should almost reach top of vegetables. Bring to a boil on the stove, and then return to oven and cook until vegetables are tender, about 30 minutes. Lift out the vegetables and arrange them on a serving platter. Transfer pork to another plate. Cover both and keep warm near the stove.

6. Make a Beurre Manie: Rub the softened butter together with the flour until completely incorporated. Pour off and measure the cooking liquid remaining in the pot; you should have about 2 cups. Return it to the pot and boil until reduced to 1 cup, about 6 minutes. Whisk in the beurre manie and continue whisking until the liquid comes to a boil, then lower heat and simmer for 1 minute (to remove the raw starchy taste). Turn off the heat and stir in the cream (if using) and mustard.

7. Serve: Use a fork to shred the meat into large chunks. Transfer to platter with vegetables. Serve with sauce and more mustard on the side.