Preheat the oven to 300°F. Pat the roast dry with towels. Generously season with salt and pepper on all sides.
In a large, oven-safe Dutch oven, heat the oil over medium-high heat until it starts to shimmer. Sear the meat on all sides until it's deeply browned. Set aside on a plate.
Add the shallots to the Dutch oven and cook until very soft (add additional oil if the pan is dry). Add the garlic and thyme, and saute until fragrant. Deglaze the pan with the wine, and let it reduce by half, about four minutes. Make sure to scrape up any browned bits on the bottom of the pan! Stir in the Dijon mustard, and place the roast back in the Dutch oven.
Add the beef broth until it comes halfway up the roast (about 3 cups). Scatter the carrots and potatoes around the roast.
Place the lid on the Dutch oven. Bake the roast for about 3 hours, until it easily shreds apart. If it's tough and chewy, place the roast back in the oven for 30-60 minutes, until it's nice and tender. If the braising liquid starts to evaporate, add more to prevent the roast from drying out.
Remove the roast from the Dutch oven, and tent with foil. Let it rest for 10-15 minutes, and then shred or slice. Arrange the meat on a platter, and scatter the potatoes and carrots around the roast. Discard the remaining sprigs of thyme, and spoon the juices over the top. Serve!