Preheat oven to 250°. Put lamb in a large (12- by 16-in.) roasting pan with fold-down handles (upright handles will make it hard to seal tightly with foil). Make several shallow slashes all over meat. Put anchovies, olives, 1 1/2 tbsp. capers, and 1 tbsp. oregano in a food processor with 1/3 cup oil and whirl into a paste. Slather paste all over lamb.
Pour wine into pan and cover tightly with foil to seal. Roast lamb 4 hours.
Meanwhile, trim artichokes: Snap off more than half of leaves (down to the yellow layer), then slice off green tips of remaining leaves and peel fibrous surface from stems. Peel shallots and separate into lobes. Scrub carrots and trim stems to 1 in.
Transfer about 1/2 cup juices from lamb to another roasting pan, using a bulb baster. Divide artichokes, shallots, and about half of thyme sprigs between pans, arranging in a single layer. Add 1/2 cup water to pan of vegetables, season with salt and pepper, and seal both pans tightly with foil.
Roast vegetables and lamb 1 hour; then add carrots to pans and reseal tightly. Roast until vegetables and lamb are fork-tender and lamb is pulling away from the bone, 1 1/2 to 2 1/2 hours more. Remove both pans from oven and increase heat to 450°.
Transfer juices from lamb and vegetables to a bowl, using bulb baster. Drizzle pans of lamb and vegetables with remaining 1/3 cup oil and roast, uncovered, until crisp and browned, about 30 minutes.
Stir remaining capers and oregano into vegetables, dividing between pans, and sprinkle with remaining thyme. Let meat rest 20 minutes.
Spoon fat off juices, then purée juices in a blender with 4 shallots. Pour sauce into a large frying pan and boil until deep brown, 5 to 6 minutes. Pour into a serving bowl.
Set pan of lamb and vegetables on table, with extra vegetables and sauce on the side. Slice meat across the grain or pull off with a serving fork.
*If packed in salt, soak in water 10 to 15 minutes, then drain.