Pat the beef dry with paper towels. Sprinkle the beef generously with salt and pepper. Heat 2 tablespoons olive oil in a heavy 6-quart roasting pan over medium-high heat. Add the beef and cook until brown on all sides, about 15 minutes total cooking time. Transfer the beef to a bowl. Add remaining tablespoon oil to the pan, add the onions and saute until tender, scraping up the brown bits on the bottom of the pot, about 5 minutes. Add the garlic and saute 1 minute. Add the wine and boil 1 minute. Stir in the broth and mushrooms. Return the beef to the pan. Bring the liquids to a boil. Cover and transfer to the oven. Braise until the beef is fork-tender, turning the beef over halfway through cooking, about 3 hours.
Transfer the beef to a cutting board. Tent the beef with foil and let stand 15 minutes. Meanwhile, spoon any excess fat off the top of the pan juices. Transfer the pan juices and vegetables to a blender and puree until smooth. Combine the sauce and rosemary sprig in heavy medium saucepan. Bring to a boil. 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 and garnish with rosemary. Spoon the sauce over and serve, passing the remaining sauce in a sauce boat.