Preheat the oven to 350F. Lay the roast on triple layers of paper towels, and dab off any excess moisture. Sprinkle generously with salt and pepper. Heat 2 tablespoons of vegetable oil to a shimmer, in a heavy, 6-quart pot or Dutch oven over medium-high heat, and add the beef and cook until browned on all sides, about 12 minutes. (Don't rush this step.) Transfer the beef to a plate and set aside.
Reduce the heat to medium. Add the onions to the same pan and saute until tender, scraping up the brown bits on the bottom of the pot, about 5 minutes. Add the garlic and saute for 1 minute. Add the wine and boil for 1 minute. Stir in the broth and mushrooms. Return the beef to the pan and bring the liquids to a boil. Cover the pot and transfer to the oven. Braise until the beef is fork-tender, about 3 hours, turning the beef over halfway through.
Transfer the beef to a cutting board. Tent with foil and let stand for 15 minutes. Meanwhile, spoon any excess fat off the top of the pan juices. Using a stick blender, puree sauce until smooth. In a medium, heavy saucepan, combine the sauce and rosemary sprig. Bring to a boil, then season the sauce to taste with salt and pepper.
Cut the beef across the grain into 1/2-inch-thick slices. Arrange the sliced beef on a platter. Spoon some of the sauce over and serve, passing the remaining sauce in a gravy boat.