Pat the oxtails dry with paper towels and season generously with salt and black pepper on all sides. Heat the olive oil in a large heavy-bottomed pot or Dutch oven over medium-high heat.

Add the oxtails in a single layer, working in batches to avoid crowding the pan. Sear for 3 to 4 minutes per side until a deep, golden-brown crust forms, then transfer the meat to a plate.

Lower the heat to medium and add the chopped onion and carrots to the remaining beef fat in the pot. Sauté for 5 to 7 minutes until the onions are softened and lightly browned.

Stir in the minced garlic and tomato paste, cooking for 1 minute until fragrant and the tomato paste turns a dark brick red. Pour in the red wine, scraping up any browned bits from the bottom of the pot with a wooden spoon.

Return the oxtails and any resting juices to the pot, pouring in the beef broth, dried thyme, and bay leaves. Bring the liquid to a gentle boil, cover tightly, and reduce the heat to low.

Simmer gently for about 1 hour and 45 minutes, checking occasionally to ensure there is enough liquid, until the meat is incredibly tender and nearly falling off the bone. Carefully skim any excess fat that has risen to the surface.

Stir in the thawed broad beans and simmer uncovered for a final 15 minutes to allow the beans to soften and the rich sauce to thicken slightly. Remove the bay leaves and serve hot with a side of crusty bread to mop up the sauce.