Preheat your oven to 325°F (160°C).

In a large oven-safe pot or Dutch oven, heat the olive oil over medium-high heat. Season the lamb leg with salt and pepper. Sear the lamb in the pot until browned on all sides, about 4-5 minutes per side. Remove the lamb and set aside.

In the same pot, add the minced garlic, fresh thyme, and rosemary. Sauté for 1-2 minutes until fragrant.

Pour 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 chicken or vegetable broth, chopped carrots, parsnips, and onion to the pot. Stir in the honey and return the lamb leg to the pot, nestling it in the vegetables and liquid.

Bring the mixture to a gentle simmer, then cover the pot and transfer it to the preheated oven. Braise for 2.5 to 3 hours, or until the lamb is tender and easily pulls apart with a fork.

Once cooked, remove the pot from the oven and carefully take out the lamb leg. Let it rest for about 10-15 minutes.

For the gravy, if you prefer a thicker consistency, mix the cornstarch with a little water to create a slurry, then stir it into the pot with the vegetables and broth. Bring to a simmer over medium heat and cook until thickened.

Slice the lamb leg and serve it with the root vegetable gravy. Garnish with fresh thyme sprigs.