Preheat the oven to 325°F (160°C).
Pat the beef cubes dry with paper towels and season them generously with salt and freshly ground black pepper.
In a large Dutch oven or heavy-bottomed ovenproof pot, heat 2 tablespoons of olive oil over medium 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.
Increase the heat to medium-high. In batches, sear the beef cubes in the bacon fat until browned on all sides. Avoid overcrowding the pot. Transfer the browned beef to a plate.
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 well to coat. Cook for 2 minutes to remove the raw flour taste.
Slowly pour in the red wine, stirring to deglaze the pot and scrape up any browned bits from the bottom.
Add the beef broth, tomato paste, fresh thyme, and bay leaves. Stir to combine.
Bring the mixture to a simmer, then cover the pot with a lid and transfer it to the preheated oven.
Cook for 2 1/2 to 3 hours, or until the beef is very tender.
While the beef is cooking, prepare the pearl onions by blanching them in boiling water for 2 minutes, then draining and peeling them.
In a large skillet, melt 1 tablespoon of butter over medium heat. Add the pearl onions and cook until they are lightly browned and tender, 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 browned and their moisture has evaporated, about 8 minutes. Remove and set aside.
When the beef is tender, remove the pot from the oven. Discard the bay leaves and thyme stems if any remain.
Stir the cooked pearl onions and mushrooms into the stew.
Simmer the stew uncovered on the stovetop over medium heat for 10 minutes to blend the flavors and thicken the sauce slightly.
Adjust seasoning with salt and pepper to taste.
Sprinkle the chopped fresh parsley over the stew before serving.