Preheat your oven to 325°F (163°C).
Pat the beef chunks dry with a paper towel and season generously with salt and black pepper.
Heat 1 tablespoon of olive oil in a large oven-safe Dutch oven over medium-high heat. Add the beef chunks in batches so as not to overcrowd the pan. Sear until deeply browned on all sides, about 6-8 minutes per batch. Remove the beef and set it aside on a plate.
Lower the heat to medium and add the remaining 1 tablespoon of olive oil. Add the leeks and carrots, sautéing for 5 minutes until they begin to soften. Stir in the minced garlic and cook for 1 more minute until fragrant.
Stir in the tomato paste and flour. Cook for 1-2 minutes, stirring constantly to lightly toast the flour.
Pour in the dry red wine to deglaze the pot, using a wooden spoon to scrape up the browned bits from the bottom. Let it bubble and reduce for about 2 minutes.
Return the seared beef and any resting juices back into the Dutch oven. Pour in the beef broth. Add the chopped mushrooms, fresh thyme sprigs, and bay leaves. Stir well to combine, bringing the liquid to a gentle simmer.
Cover the Dutch oven with a tight-fitting lid and carefully transfer it to the preheated oven. Braise for 2 to 2.5 hours, or until the beef is completely fork-tender.
Remove the pot from the oven. Discard the bay leaves and thyme stems. Taste the sauce and adjust the salt and pepper if needed. Serve immediately over mashed potatoes or egg noodles.