Into a bowl of water, squeeze the juice from 2 lemon quarters, dropping squeezed quarters into water. Break off and discard the stem from 1 artichoke. Bend back the outer leaves until they snap off close to the base and remove several more layers of the leaves until the pale inner leaves are reached. Trim the base and the sides of artichoke with a very sharp stainless-steel knife and cut off the top 1½ inches. Cut the artichoke into quarters and cut away the choke and spiky purple-tipped leaves. Rub the artichoke quarters all over with a remaining lemon quarter and drop it in the bowl of water. Prepare the other 3 artichokes in the same manner, using other lemon quarter if necessary.
Preheat the oven to 475*F.
In a saucepan of boiling salted water, blanch the artichokes for 3 minutes and, with a slotted spoon, transfer to a bowl. In boiling salted water, blanch the Jerusalem artichoke for 5 minutes, and drain.
In a flameproof roasting pan, 17 by 11½ by 2 inches, toss the artichokes, the Jerusalem artichokes, onions, and garlic cloves with 2 tbsps olive oil and salt and pepper to taste. Roast the vegetables in middle of oven for 20 minutes.
While the vegetables are roasting, heat a heavy 12-inch skillet over moderately-high heat until hot. While skillet is heating, season lamb racks with salt and pepper. Add vegetable oil to the skillet and brown the lamb, meaty sides down, for 2 minutes. Turn the lamb over and brown for 2 minutes more. Transfer lamb to a plate and cool slightly.
In a small bowl, stir together the chopped garlic, oregano, coarse salt, freshly ground pepper, and remaining 2 tsps of olive oil, and rub all over the lamb.
Remove the pan from the oven and stir the vegetables. Arrange the lamb over the vegetables, rib sides down, and roast for 20 minutes. or until a meat thermometer inserted in the fleshy section registers 130°F for medium-rare.
Remove the 8 garlic cloves from the pan and squeeze the garlic out of skins into a blender. Add broth and blend until smooth. Transfer the remaining vegetables and the lamb to a heated platter, reserving the juices in the pan, and let rest, covered loosely for 10 minutes.
To the pan, add the garlic broth and salt and pepper to taste, and on top of the stove, deglaze over moderately-high heat, scraping up the brown bits. Boil the sauce until it thickens slightly and transfer to a sauceboat.
Cut the lamb racks in half or into individual chops and serve with vegetables.