Heat the oven to 150C (non-fan 170C). Season the lamb shanks with sea salt and freshly ground black pepper. Place a large casserole dish on a medium heat and add the vegetable oil, then the lamb. Brown the lamb all over for 8–10 minutes, then remove from the casserole dish onto a tray.
Meanwhile, cut the carrots, shallots and onions into 2½cm pieces, keeping them separate. Add the butter to the oil in the casserole dish, and when it melts, add the carrot. Start to colour this up, then after5 minutes add the split garlic cloves and thyme. Keep colouring these for a few more minutes, then add the shallot, onion, 1 tsp of sea salt and 4–6 turns of black pepper.
After another 5–8 minutes, when the vegetables are all coloured, turn the heat down and add the tomato paste. Cook for a further 2 minutes, then add the flour and cook for 2 more minutes.
Add the stock and tomatoes, turn the heat up and bring this to a slow simmer, then skim off the scum. Add the pearl barley and return the lamb to the casserole dish. Cover with a lid and cook in the oven for 2½ hours.
While this is cooking, you can make your honey-roast garlic. Place a small sauté pan on a very low heat. Add the olive oil and butter and, when the butter has just melted, add the garlic, cut-side down, with the thyme.
Add a little salt and cook very slowly until the garlic starts to colour. After approximately 15 minutes, add the honey, then continue cooking until the garlic is soft (about another 15–20 minutes).
Once the lamb is cooked, serve a shank on each plate with some of the pearl barley and vegetables, and a half garlic bulb.