Preheat the oven to 325°F (160°C).
Pat the beef cubes dry with paper towels and season generously with salt and freshly ground black pepper.
In a large Dutch oven or heavy ovenproof pot, heat 2 tablespoons of olive oil over medium-high heat.
Add the diced bacon and cook until crisp. Remove the bacon with a slotted spoon and set aside, leaving the fat in the pot.
Add the beef cubes in batches to the pot, browning them on all sides. Avoid overcrowding. Remove browned beef and set aside.
Add the remaining 1 tablespoon of olive oil to the pot if needed. Add the sliced carrot and chopped onion and cook until softened, about 5 minutes.
Add the minced garlic and cook for 1 minute until fragrant.
Return the beef and bacon to the pot. Sprinkle the flour over the meat and vegetables and stir to coat evenly. Cook for 2 minutes to remove the raw flour taste.
Slowly pour in the red wine, scraping the bottom of the pot to loosen browned bits.
Add the beef broth, tomato paste, fresh thyme, and bay leaf. Stir to combine.
Bring the mixture to a simmer, then cover the pot and transfer it to the preheated oven.
Cook in the oven for 2 1/2 to 3 hours, or until the beef is very tender.
While the beef cooks, prepare the pearl onions and mushrooms: In a skillet, melt 1 tablespoon of butter over medium heat. Add the pearl onions and sauté until they begin to brown and soften, about 10 minutes. Remove and set aside.
In the same skillet, melt the remaining 1 tablespoon of butter. Add the quartered mushrooms and sauté until golden brown, about 8 minutes. Remove and set aside.
When the beef is tender, remove the pot from the oven and place it on the stove over medium heat.
Discard the bay leaf and thyme sprigs.
Add the sautéed pearl onions and mushrooms to the stew and simmer uncovered for 10 minutes to blend flavors and thicken the sauce slightly.
Adjust seasoning with salt and pepper to taste.
Sprinkle chopped fresh parsley over the stew before serving.