Heat a large skillet over medium-high heat. Cook the bacon until golden and crispy and the fat has rendered. Transfer to a paper towel-lined plate and set aside. Pour off (and reserve!) all but a tablespoon of the bacon fat from the skillet.
Pat the beef dry and season with salt and pepper. Return pan to medium-high heat, and when the fat is shimmering and there is a wisp or two of smoke, arrange a single layer of meat in the pan, being sure not to crowd the pan. Sear without moving it, until the beef releases easily from the pan and the underside is golden-brown, 1-3 minutes. Turn over and sear the other side the same way. As you work, transfer the beef to a slow-cooker (if using -- see note) or to a Dutch oven or heavy, 6-quart, covered baking dish. After each batch of meat, pour 1/4 cup of the wine into the hot pan, and deglaze over medium-high heat. As the wine simmers, scrape up the crispy bits and the dark glaze from the bottom of the pan. Simmer for 1-2 minutes, then pour the wine over the meat. Before placing the next layer of meat in the skillet, melt a tablespoon of reserved bacon fat in the pan. Finish by pouring the wine and de-glazed bits over the meat so you have a clean pan.
After you've seared all the meat, add one more tablespoon of reserved bacon fat to the ban and reduce the heat to medium. Add the onions and a generous pinch of salt, and cook 6-8 minutes, until soft and browned. Stir in the carrots and celery, and cook until softened. Stir in the garlic and tomato paste, and cook 1-2 minutes, until fragrant. Transfer the vegetable mixture to the slow-cooker (if using) or to the Dutch oven. If using the oven, turn it on now to 300˚F.
With a paper towel, wipe the pan clean, and if some is remaining, add one more tablespoon of reserved bacon fat to the pan (if not, use vegetable oil). Add the mushrooms and a pinch of salt, and cook over medium heat for 8-10 minutes, until they have released their liquid, the liquid has evaporated, and the mushrooms are golden brown. Transfer the mushrooms to a small bowl.
If using slow-cooker: Sprinkle the beef and vegetable mixture with 1 teaspoon of salt, and stir. Add the thyme sprigs and bay leaf, then pour in remaining wine and broth (the liquid will not quite cover the mixture -- it should only come up about 3/4 of the way). Cover, and cook on LOW for 6-8 hours, until the beef is very tender.
If using the oven: Add 1 teaspoon of salt to the beef and vegetable mixture and stir. Add the thyme sprigs and bay leaf, then pour in remaining wine and broth (the liquid will not quite cover the mixture). Cover and place in the preheated oven. Cook for 2 hours, then check the meat for doneness every 15 minutes. The beef should be very tender, and exact cooking times can vary.
Either method: When the meat is cooked, fold in the reserved bacon and the mushrooms. With the slow cooker, cook on HIGH for 10 more minutes. With the oven method, move the Dutch oven to the stovetop and simmer over medium heat for 10 more minutes.
To serve, sprinkle with chopped parsley. Pour yourself a glass of wine, and enjoy!