Season the beef brisket generously with salt and pepper on all sides.

In a large, heavy-bottomed pot or Dutch oven, heat olive oil over medium-high heat. Add the brisket and sear for about 5-7 minutes on each side until browned. Remove the brisket from the pot and set aside.

In the same pot, add the minced garlic and chopped shallots. Sauté for about 2-3 minutes until fragrant and the shallots are translucent.

Pour in the balsamic vinegar, scraping the bottom of the pot to release any browned bits. Allow it to simmer for about 2 minutes.

Stir in the beef broth, Dijon mustard, brown sugar, and tarragon. Mix well to combine.

Place the seared brisket back into the pot, ensuring it is submerged in the liquid.

Arrange the chopped carrots, parsnips, and potatoes around the brisket. Season the vegetables with additional salt and pepper.

Cover the pot with a lid and transfer it to a preheated oven at 325°F (163°C). Braise for about 3 to 4 hours, or until the brisket is tender and easily shredded with a fork.

Once the brisket is cooked, remove it from the pot and let it rest for about 15 minutes. Meanwhile, strain the braising liquid into a saucepan and bring to a simmer. Reduce the liquid for about 10 minutes until thickened slightly. Adjust seasoning if necessary.

Slice the brisket against the grain and serve with the root vegetables and rich gravy. Garnish with fresh tarragon sprigs if desired.