Start by preparing the beef and vegetables. Cut the beef chuck into bite-sized cubes and season with salt and pepper. Chop the carrot and onion into small pieces, and mince the garlic.

Heat the olive oil in a large pan over medium-high heat. Sear the beef pieces until browned on all sides, about 5 minutes. Remove the beef and set aside.

In the same pan, add the butter and melt it. Add the chopped onion, carrot, and minced garlic. Sauté for 3-4 minutes until the onion becomes translucent.

Stir in the tomato paste and cook for another minute. Add the beef back into the pan along with the quartered mushrooms.

Pour in the red wine, scraping the bottom of the pan to deglaze it. Allow the wine to reduce for about 5 minutes.

Add the beef stock and fresh thyme. Bring to a simmer, then cover and let it cook on low heat for 45 minutes or until the beef is tender.

While the beef is cooking, peel and chop the russet potatoes into equal-sized chunks. Add them to a pot of cold salted water and bring to a boil. Cook until the potatoes are fork-tender, about 15 minutes.

Drain the potatoes and return them to the pot. Add the milk and a little more butter, and mash until smooth and creamy. Season with salt and pepper to taste.

Once the beef is tender, adjust seasoning as needed with salt and pepper.

Serve the Beef & Mushroom Bourguignon over the creamy mashed potatoes, garnished with a sprig of fresh thyme if desired.