Preheat your oven to 350°F (175°C).
In a large Dutch oven or heavy pot, heat the vegetable oil over medium-high heat. Season the oxtails with salt and pepper, then brown them in batches until golden on all sides. Remove the oxtails and set aside.
In the same pot, add the chopped onion, garlic, carrots, and celery. Sauté for about 5 minutes, or until the vegetables are softened.
Stir in the tomato paste, cooking for another minute.
Pour in the red wine, scraping the bottom of the pot to deglaze. Allow it to simmer for 2-3 minutes.
Return the oxtails to the pot, then add the beef broth, prunes, thyme, and bay leaf. Bring the mixture to a gentle boil.
Cover the pot with a lid and transfer it to the preheated oven. Braise for 2.5 to 3 hours, or until the oxtails are tender and falling off the bone.
Once done, remove the pot from the oven. Discard the bay leaf and adjust seasoning with salt and pepper if needed.