Pour olive oil into dutch oven and fry the oxtails until browned on both sides. Finely chop the onion and 1 carrot. Remove the oxtails and fry the vegetables for 2-3 minutes.
Put the oxtails back into the pot. Add water or broth so that the liquid covers the meat so they are submerged. Add thyme, bay leaf, garlic, black pepper and salt. Boil covered over medium heat for 3 hours.
If you want to add vegetables into the pot together: Prep veggies by cutting into bite size pieces, could add 1-2 regular potatoes as well. Add in 30 minutes before meat is done, so after 2.5 hrs of boiling. Cover again and continue cooking until veggies are tender.
The meat should come off the bone easily at this point. Strain the broth through a sieve and add the oxtails back into the pot.
For crispy veggies: Put coarsely chopped peppers, sweet potatoes and 2 carrots on a baking sheet. Cook in oven preheated to 400 F for about 10-15 minutes. Serve with oxtails.