Cut the beef chuck into bite-sized pieces, season with salt and pepper. Heat half of the olive oil in a large pot or Dutch oven over medium-high heat, and brown the beef in batches. Remove and set aside.

In the same pot, add the remaining olive oil. Add pearl onions and chopped garlic, stirring until the onions are caramelized and garlic is fragrant, about 3-4 minutes.

Add sliced cremini mushrooms to the pot, and cook until softened, about 5 minutes.

Return the beef to the pot. Sprinkle with thyme and add the bay leaf. Pour in the red wine to deglaze the pot, scraping the browned bits from the bottom with a wooden spoon.

Add the beef stock and bring to a simmer. Lower heat, cover, and let cook on low heat for about 1 hour, or until the beef is tender.

While the beef is cooking, peel and chop the carrots into bite-sized pieces. In another saucepan, add a small drizzle of olive oil, and sauté the carrots over medium heat until they start to soften, about 5-7 minutes.

Add a splash of water, cover the pan, and let the carrots braise on low heat until tender, about 15 minutes.

For the mashed potatoes, peel and chop the potatoes into even chunks. Boil in salted water until tender, about 15 minutes. Drain and return to the pot.

Mash the potatoes with 1 tbsp of butter and 2 tbsp of milk. Season with salt and pepper to taste. Keep warm.

Check the bourguignon for seasoning, adjust with salt and pepper as needed.

Serve the beef bourguignon hot with creamy mashed potatoes and braised carrots on the side. Garnish with a sprinkle of fresh thyme if desired.