Preheat the oven to 250 degrees F.
On the stovetop, brown the lamp, fat-side down, in a large roasting pan. Remove when nicely browned across its middle (you won't get much more than this) and set aside while you fry the shallots, garlic and carrot briefly. Just tip them into the pan - you won't need to add any more fat - and cook them, sprinkled with the salt, gently for a couple of minutes. Pour the water over and then replace the lamb, this time fat side up. Let the liquid in the pan come to a boil, then tent with foil and put in the preheated oven.
Now just leave it there while you sleep. I find that if I put the lamb in before I go to bed, it's perfect by lunchtime the next day. But the point is, at this temperature, nothing's going to go wrong with the lamb if you cook it for a little less or a little more.
If you want to cook the lamb the day you're going to eat it, heat the oven to 325 degrees F and give it 5 hours or so. The point is to find a way of cooking that suits you: you know what sort of pottering relaxes you and what makes you feel constrained; how much time you've got, and how you want to use it. Don't let the food, the kitchen or the imagined expectations of other people bully you.
With the homily over, about 1 hour before you want to eat, remove the lamb from the pan to a large plate or carving board - not that it needs carving; the deal here is that it's unfashionably overcooked, falling to tender shreds a the touch of a fork. This is the best way to deal with shoulder of lamb: it's cheaper than leg, and the flavor it deeper, better, truer, but even good carvers, which I most definitely am not, can get unstuck trying to slice it.
To finish the lamb salad, simply pull it into pieces with a couple of forks on a large plate. Sprinkle with more sea salt and some freshly chopped mint, then cut the pomegranate in 1/2 and dot with the seeds from 1 of the halves. This is easily done; there's a simple trick, which means you never have to think of winkling out the jeweled pips with a safety pin ever again. Simply hold the pomegranate 1/2 above the plate, take a wooden spoon and start bashing the curved skin side with it. Nothing will happen for a few seconds, but have faith. In a short while the glassy red, juicy beads will start raining down.
Take the other 1/2 and squeeze the preposterously pink juices over the warm shredded meat. Take to the table and serve.
What I do with the leftovers is warm a pita bread in the microwave, and then spread it with a greedy dollop of hummus, then take the chill off the refrigerated lamb in the microwave and stuff the already gooey pita with it. Add freshly chopped mint, black pepper and whatever else you like; raw, finely chopped red onion goes dangerously well.