Preheat oven to 300ºF. Season roast generously with salt and pepper on both sides. Heat a lidded Dutch oven or heavy pot large enough to hold the roast over medium-high heat, adding enough oil to coat the bottom. Brown the roast on both sides about 4 minutes. When you turn the roast, reduce heat to medium.
Transfer roast to a plate and add onion to the Dutch oven, drizzling in more oil, if needed. Cook, stirring occasionally, for 3 or so minutes, until it’s beginning to soften. Be careful not to brown it. Add garlic and rosemary to the Dutch oven and cook, stirring constantly, for 45 seconds or so.
Add broth, wine and mustard to the Dutch oven, stirring to combine everything and scraping up any browned bits. Return the roast to the Dutch oven—the liquid should come about halfway up the sides of the roast.
Cover Dutch oven with the lid and put it in the oven. Roast for about 2 hours, turning the roast over in the last 1/2 hour.
Turn the roast over again. Increase the oven temperature to 350ºF. Add potatoes, carrots and mushrooms to the Dutch oven, arranging them around the sides of the roast. Add a little more liquid, if needed, but the vegetables don’t need to be submerged.
Cover the Dutch oven and return it to the oven. Roast for another 1-1/2 hours. The meat and vegetables should be completely tender. Transfer the roast to a platter and serve with the vegetables.