In a large Dutch oven, heat a generous drizzle of olive oil over medium-high heat. Once shimmering, add beef chuck cubes in batches, ensuring not to overcrowd the pot. Sear each side for about 3-4 minutes until beautifully browned. Once browned, transfer the beef to a plate and set aside.

Reduce the heat to medium; add more olive oil if necessary. Toss in the chopped yellow onions, sautéing for 5-7 minutes until they turn golden and fragrant. Incorporate minced garlic and sauté for an additional minute.

Once the onion and garlic are softened, stir in sliced carrots. Cook for 2-3 minutes until they soften slightly, stirring occasionally. Then, add tomato paste, mixing thoroughly and allowing it to caramelize for another 2 minutes.

Sprinkle all-purpose flour over the sautéed vegetables, stirring quickly to coat. Gradually pour in beef stock and water, scraping up any browned bits from the pan. Return the seared beef to the pot and bring to a gentle simmer.

Add thyme, a bay leaf, and season with salt and black pepper to taste. Cover the pot and reduce the heat to low, letting the Beef Bourguignon simmer for 2.5-3 hours.

While the stew simmers, heat a skillet over medium heat and add olive oil. Sauté sliced mushrooms for about 5 minutes until they turn golden brown. Add pearl onions, cooking for an additional 5 minutes until soft.

After the stew has finished simmering, stir in the sautéed mushrooms and pearl onions. Allow everything to simmer together for another 10 minutes. Once ready, remove the bay leaf and serve warm.