Heat a generous tablespoon of olive oil in a large, heavy-bottomed pot over medium-high heat. Brown the beef chuck in batches for about 5-7 minutes per side until it develops a rich brown crust. Remove the beef and set it aside.

In the same pot, add the chopped onion and sliced carrots. Sauté over medium heat for about 5 minutes until softened.

Stir in the minced garlic, cooking for approximately 1 minute until fragrant. Add the tomato paste and continue cooking for another 2-3 minutes.

Sprinkle all-purpose flour over the sautéed vegetable mix, stirring well to combine. Cook for an additional minute.

Gradually pour in the red wine, stirring to scrape up any brown bits from the pot. Let it simmer for about 2-3 minutes.

Pour in the beef broth, along with dried thyme and bay leaves. Mix everything together and bring to a gentle simmer.

Return the browned beef to the pot, ensuring it's submerged in the liquid. Cover and reduce heat to low, allowing it to simmer gently for 1.5 to 2 hours.

In a separate skillet, melt a tablespoon of butter over medium heat. Sauté the pearl onions for about 5 minutes, then add the sliced mushrooms and cook for an additional 5 minutes.

Stir in the sautéed pearl onions and mushrooms into the pot with the stew. Simmer uncovered for another 15-20 minutes to thicken the sauce.

Adjust seasoning with salt and black pepper to taste. Remove bay leaves before serving.