Preheat the oven to 300˚F. Drizzle some oil into a Dutch oven or ovenproof braising pan over medium-high heat and brown the beef in batches, seasoning with salt and pepper. Transfer to a plate as it browns. Add the onion and garlic to the skillet and cook for a few minutes, then return the beef to the pan, pour in the wine and add enough stock to come halfway up the meat mixture. Add the tomato paste and thyme.
Cover and bake for 2 1/2 to 3 hours, until the meat is very tender. Toward the end of the cooking time, heat the butter in a skillet and sauté the mushrooms for 5-8 minutes, until golden. Stir into the beef mixture just before serving.