Cut the meat and dry it with paper towels. Add the cooking oil to a skillet and heat it over medium heat until hot. Add the beef, and brown on all sides. Do not overcrowd the pan.
Transfer the browned beef to a Dutch oven or other heavy pan with a lid.
When all the beef is browned, remove all the oil from the pan, leaving only one tablespoon. If it is burned, discard and add 1 tablespoon of fresh oil.
Add the carrots and saute until lightly browned. Then, add them to the beef in the Dutch oven.
Pour one cup of wine into the pan and deglaze it. Then, pour it into the Dutch oven.
Add the garlic, the remaining red wine, beef stock, tomatoes, bay leaf, thyme, and a sprinkling of salt.
Cover with a lid and simmer over low heat for 2 1/2 to 3 hours, or until the beef is fork-tender. Prepare the mushrooms, onions, and the buerre manié (See preparation instructions in the sections below.)
Remove the beef to another pan. Strain the sauce, pressing the residue with a spoon to extract as much liquid as possible. Spoon off as much grease as possible. Taste and season with salt if needed.
Return the beef and sauce to the stove and heat. Add the beurre manié and whisk until dissolved. Then, bring the sauce to a boil, and the butter/flour mixture will thicken it. Add the onions, mushrooms, and their juices, and cook for 2 minutes to blend the flavors.