Cut the celery and onion into long slices, and the carrot into round slices.
Dredge the oxtail in the flour so that it is coated on all sides.
Set on a plate, then heat the olive oil in a Dutch oven or large, heavy pot over medium high heat.
Add the oxtail pieces to the hot oil and cook to brown on all sides. Adjust the heat as needed, then remove meat from the pot and set aside.
Keeping the pot over medium high heat, add the vegetables and sauté, stirring occasionally. Fry until golden brown, then add oxtail back into the pot.
Lower the heat to medium and add the stock (you can add more or less to your liking, but note the end result of the oxtail soup will be more thin or more thick.)
Add the thyme, bay leaf and salt, and slowly bring to a boil. Reduce the heat upon boiling and simmer gently, covered with a lid, for at least 3 hours.
Remove from heat and allow to cool. You can refrigerate for a few hours or overnight to make this step easier. Remove the fat from the top of the oxtail soup and discard (you can reserve some to make the brown roux, if desired.)If the soup is cold, heat it slightly, so that the meat can be removed easily, place on a plate. Using your fingers, remove all the meat from the tail bones. Discard any fat. Strain the soup through a sieve into another pot. Using a spoon, take out some of the carrot rounds, and set aside along with the oxtail meat. Keep the rest of the vegetables aside for later use.
In a small pot, melt the butter and/or beef fat reserved from the soup, then add the flour and stir over medium to medium high heat. Continue to cook until the mixture turns a golden brown color.
Keeping the oxtail soup over low heat, add a little of the strained oxtail broth to the brown roux, and stir quickly. The mixture will absorb the liquid, then add a little more and continue until the roux is runny. This ensure no lumps in your oxtail soup.
Pour the roux into the soup and stir.
Place some of the reserved vegetables into a glass jug or deep bowl. The amount you use is up to you (I used about half of the veg, more of the onion and some carrots.)
Pour some of the oxtail broth into the jug so that it is well covered. Blend using an immersion blender until smooth (alternatively, you can do this in a small food processor or blender-be careful if the liquid is hot.)
Pour the blended vegetable mixture into the pot and stir.
Next add the reserved carrots and oxtail meat.
Using a small, fine sieve, remove the little bits of fat which will float to the top of the oxtail soup, and discard. Heat to boiling, taste for salt and pepper, adjust as desired, then simmer for 2 minutes.
Ladle into serving bowls, being sure to scoop some of the oxtail meat into each bowl. Sprinkle with freshly chopped parsley and serve with fresh, crusty bread.