Preheat your oven to 325°F (163°C).
In a large, oven-safe pot or Dutch oven, heat the olive oil over medium-high heat. Season the lamb shanks with salt and pepper, then sear them in the pot until browned on all sides, about 8-10 minutes. Remove the shanks and set aside.
In the same pot, add the diced onion, carrots, and celery. Sauté for about 5 minutes until the vegetables are softened. Add the minced garlic and cook for an additional minute.
Pour in the stout beer, scraping the bottom of the pot to deglaze. Bring to a simmer and let it reduce for about 5 minutes.
Stir in the beef broth, tomato paste, thyme, rosemary, bay leaves, Worcestershire sauce, and balsamic vinegar. Return the lamb shanks to the pot, ensuring they are submerged in the liquid.
Cover the pot with a lid and transfer it to the preheated oven. Braise for 2.5 to 3 hours, or until the meat is tender and falling off the bone.
Once done, remove the pot from the oven. Discard the bay leaves and skim any excess fat from the surface.
Serve the lamb shanks with the braising liquid and vegetables spooned over the top. Garnish with chopped parsley.