1. Pat the beef cheeks dry with paper towels and season generously with salt and pepper on all sides.

2. Heat the olive oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Add the beef cheeks in a single layer (work in batches if necessary) and sear until deeply browned on all sides, about 4-5 minutes per side. Remove the beef cheeks from the pot and set aside.

3. Reduce the heat to medium. Add the chopped onion, carrots, and celery to the pot. Sauté for 8-10 minutes, until softened and lightly caramelized. Add the minced garlic and cook for another minute until fragrant.

4. Stir in the tomato paste and cook for 1 minute. Pour in the red wine, scraping up any browned bits from the bottom of the pot. Bring to a simmer and let it cook for 5 minutes, allowing the wine to reduce slightly.

5. Return the seared beef cheeks to the pot. Add the beef broth, thyme sprigs, and bay leaves. The liquid should almost completely cover the meat.

6. Bring the mixture to a simmer, then reduce the heat to low, cover the pot, and let it cook gently for 3.5 to 4 hours, or until the beef cheeks are fall-apart tender. Alternatively, you can cook it in an oven preheated to 325°F (160°C) for the same amount of time.

7. Carefully remove the tender beef cheeks from the pot. Strain the braising liquid through a fine-mesh sieve into a saucepan, discarding the solids. Bring the liquid to a boil over medium-high heat and let it reduce until it has thickened into a glossy sauce, about 10-15 minutes.

8. Serve the beef cheeks whole or shredded, generously spooning the sauce over the top. Garnish with fresh parsley. It pairs wonderfully with mashed potatoes or creamy polenta.