Preheat the oven to 180C/160 fan/Gas 4.
Heat the oil over a medium heat in a large roasting tin on the hob. Add the short ribs and fry for 3-4 minutes on all sides, or until browned all over. Add the onions and carrots, then transfer to the oven for 45 minutes.
After 45 minutes, tuck the tomatoes and the whole garlic bulb into the roasting tin, pour in the beef stock and continue to cook for a further hour.
Towards the end of the cooking time, cook the spaghetti according to the packet instructions, then drain well and set aside.
Remove the beef ribs from the roasting tin and set aside on a chopping board to rest.
Squeeze the garlic cloves from their skins into the roasting tin. Using a slotted spoon or fork, lightly squash the tomatoes and mix them and the garlic into the sauce.
Slide the short-rib meat off the bones and cut it into strips. Return the meat to the roasting tin.
Stir in the cooked pasta, then season with salt and freshly ground black pepper, and serve.