Reduce the heat to medium. Add the remaining oil and the onions. Cook, stirring frequently, until tender, about 8 minutes. Add the garlic and cook for 1 minute until aromatic. Add the wine and scrape up the brown bits that cling to the bottom of the pan with a wooden spoon. Stir in the broth and mushrooms. Return the beef to the pot and bring the liquid to a boil. Cover the pot and transfer to the oven. Cook until the beef is fork-tender, about 3 hours, turning the beef over halfway through and adding more beef broth, as needed.