Season the beef with salt and pepper. Sprinkle with 2 tablespoons of the flour and toss to evenly coat.
Heat the olive oil in a large Dutch oven over medium heat. Add the bacon and cook, stirring occasionally, until browned, 6 to 8 minutes. Remove with a slotted spoon and set aside. Drain all but 1 tablespoon of the bacon fat from the pot. Add the beef and brown in small batches, 2 to 3 minutes per side. Remove the beef from the pot and set aside.
Add the carrots, onions and celery and season with 2 teaspoons each salt and pepper. Cook, stirring occasionally, until the onions are translucent, about 10 minutes. Stir in the garlic and cook an additional 1 minute. Add the bay leaves. Stir in the tomato paste and cook for 2 minutes. Add the pearl onions, thyme and reserved bacon and beef. Add the wine plus 1 to 2 cups of water (enough to cover the beef) and bring to a simmer. Simmer, covered, until the beef is very tender, about 1 1/2 hours.
Add the mushrooms to the stew. In a small bowl, use a fork to combine the softened butter with the remaining 2 tablespoons flour. Push the flour mixture through a strainer into the stew. Bring to a boil, then reduce to a low simmer. Cook until the broth is slightly thickened, about 20 minutes.
Ladle into individual bowls over mashed potatoes and serve.
This recipe has been updated and may differ from what was originally published or broadcast.