First, chop the leftover prime rib into 1 inch chunks then set to one side.
Peel & finely dice the onions & carrots. Dice the celery as well but there's no need to peel - just make sure they're clean. Grate the garlic now as well but set to one side.
Next, set a large saucepan over a medium-high heat, add in the vegetable oil then leave to heat up for a minute or two. Once hot, add in the chunks of prime rib then cook for 3-4 minutes, stirring often until well seared.Make sure not to overcrowd the pan. If needed, sear the beef in batches. Once seared, use a slotted spoon to transfer the beef to a bowl, leaving the rendered fat in the pan.
Turn the heat down to medium then add in the diced onion, carrot & celery, along with a pinch of salt then cook for 4-5 minutes until soft. Make sure to stir the veg frequently. When the veg is cooked, add in the grated garlic then cook for another 30 seconds.
Next, stir in the tomato paste, cook for a minute then pour in the red wine, scraping the bottom of the pan with a wooden spoon/spatula to deglaze.
Let the red wine reduce by about two thirds then add in the beef stock, tin of tomatoes & herbs. Add the seared beef back in then season the sauce with salt & pepper.
Turn the heat down to medium-low, cover the pan with a lid then gently simmer the ragù for 2-2½ hours until the beef is tender. You need to be able to shred it easily.To test the beef, lift a piece out of the sauce then use 2 forks to see if you can pull it apart. Once the beef is ready, use a slotted spoon to lift it out of the sauce & onto a tray. Set this aside for now. Take the herbs out of the sauce now as well; they can be discarded!