Preheat your oven to 300°F (150°C).

In a large heavy-bottomed pot or Dutch oven, heat the olive oil over medium-high heat. Season the lamb shoulder generously with salt and pepper. Once the oil is hot, sear the lamb on all sides until browned, about 4-5 minutes per side. Remove the lamb and set aside.

In the same pot, add the chopped onion and garlic. Sauté until the onion becomes translucent, about 5 minutes.

Stir in the balsamic vinegar and scrape up any browned bits from the bottom of the pot. Allow it to simmer for 2-3 minutes.

Add the beef broth, chopped tarragon, diced carrots, diced parsnips, diced potato, tomato paste, and thyme. Stir to combine.

Return the lamb shoulder to the pot, nestling it into the vegetable mixture. Bring to a gentle simmer.

Cover the pot with a lid and transfer it to the preheated oven. Braise the lamb for 4-5 hours, or until the meat is tender and easily pulls away from the bone.

Once cooked, remove the pot from the oven. Carefully take out the lamb and let it rest for 15 minutes before slicing.

While the lamb rests, place the pot with the gravy on the stovetop over medium heat. Simmer for 10-15 minutes to thicken, adjusting seasoning with salt and pepper if needed.

Serve the sliced lamb shoulder drizzled with the rich root vegetable gravy and garnish with fresh tarragon leaves.