Season the beef cheeks generously with salt and pepper on both sides.

In a large, heavy-bottomed pot or Dutch oven, heat the olive oil over medium-high heat. Add the beef cheeks and sear for about 4-5 minutes on each side until browned. Remove the meat from the pot and set aside.

In the same pot, add the chopped onion and sauté for about 3-4 minutes until translucent. Add the minced garlic and cook for another minute until fragrant.

Pour in the pomegranate juice and balsamic vinegar, scraping up any browned bits from the bottom of the pot. Stir in the honey and fresh thyme.

Return the beef cheeks to the pot, add the beef stock, and bring to a simmer. Cover the pot and reduce the heat to low. Let it braise for 2-3 hours, or until the beef is tender and easily pulls apart with a fork.

In a separate pot, bring water to a boil and add the chopped root vegetables. Cook for about 15-20 minutes, or until tender. Drain and set aside.

Once the beef is cooked, remove it from the pot and set aside. In the pot, add the butter and flour to create a roux, cooking for 1-2 minutes until golden. Gradually whisk in the braising liquid, cooking until the gravy thickens. Season with additional salt and pepper to taste.

Slice the beef cheeks and serve them over the cooked root vegetables. Drizzle with the velvety root vegetable gravy and garnish with fresh pomegranate seeds.