Preheat your oven to 350 F.
Heat the lard, tallow, or butter in a pan and brown the oxtail pieces on each side.
Remove the oxtail and brown the vegetables for the main braising liquid (carrots, celery, onions, and leeks). Make sure you stir and scrape off any meat residue in the bottom of the pan.
Add the tomatoes, thyme, bay leaves, and garlic, and cook for a couple of minutes.
Put the oxtail and browned vegetables in an ovenproof casserole and add the wine, if using.
Boil the wine at high heat until almost evaporated, and add enough stock to cover the oxtail entirely.
Bring the whole preparation to a simmer, then cover and put it in the oven to braise for about 1.5 hours to 2 hours.
At this point, most chefs will strain the liquid to remove the small pieces of cooked and mushy vegetables, so it looks better. I personally like to leave them in, but it’s up to you. You should definitely remove the bay leaves, though. Even cooking for several hours will not soften a bay leaf. We remove them because the edges can be sharp and can scratch your throat, stomach, or intestines if inadvertently swallowed.
Blanch the garnishing vegetables other than the tomatoes in boiling water for about 4 minutes.
Put the blanched vegetables in the oxtail preparation and simmer for about 2 minutes.
Enjoy this sophisticated but simple dish with family and friends!