Make the Artichokes:
Trim away the outer tough skin on the artichoke stems with a sharp paring knife. Tear away the tough outer leaves, down to the pale yellow center, then trim the tops off the pale leaves. As you are working, put the trimmed artichokes in a bowl of water with the juice and halves of 1 of the lemons. In a sauté pan with steep sides, combine the 1/4 cup oil and the garlic and sauté over medium-high heat until starting to brown, about 3 minutes. Add the artichokes and sauté until lightly browned on all sides. Add the thyme sprigs, wine, and broth and bring to a boil. Reserve 1 of the lemon halves and squeeze the rest into the cooking liquid; season lightly with salt and pepper. Reduce the heat and simmer until tender, about 20 minutes. A wooden skewer inserted into an artichoke should slide in and out smoothly. Remove and discard the thyme sprigs. Remove 6 of the nicest artichokes, cut in half, and reserve for garnish. Add the thyme leaves to the sauté pan and continue to boil the artichokes in the braising liquid until very soft, 5 to 8 minutes. Put a fine-mesh sieve over a small saucepan and pour in the braised artichokes and their liquid, saving the liquid in the saucepan. Boil the liquid over high heat until reduced by half; set aside. Transfer the braised artichokes to a blender, along with a few tablespoons of the cooking liquid, and purée, adding more liquid a little at a time to make a very smooth purée. Push the purée through the sieve and return it to the blender. Add the butter and purée again. Season with salt and pepper to taste and more lemon juice if necessary. Cover to keep warm, and set aside.

Make the Rockfish:
Preheat the oven to 375°F. Sprinkle each fillet lightly with salt and pepper; dip the top side in the buttermilk and then the bread crumbs. In a large, ovenproof sauté pan, heat the vegetable oil over medium-high heat. Carefully place the fillets in the pan, breaded side down. Cook until browned and crisp on the bottom, about 3 minutes, then turn the fillets over and put the pan in the oven for about 3 minutes, until cooked through. In a medium sauté pan, heat 1 tablespoon of the olive oil over medium heat and add the reserved artichoke halves, cut side down. Cook until browned, about 3 minutes, then turn over and cook for 1 minute. Remove from the pan and set aside. Add the remaining 1 tablespoon olive oil to the pan, along with the dried tomato halves, and cook until warmed through. Mound the artichoke purée in the center of each of 4 warmed serving plates. Top with a rockfish fillet. Garnish each plate with 3 artichoke halves and 2 tomato halves. In a small bowl, toss the olives with the preserved lemon, a drizzle of olive oil, and the capers and add some of the mixture to each plate. Garnish with basil sprigs and serve immediately.

NOTE: Preserved lemons can be found in jars in Moroccan and Middle Eastern markets. You can substitute seeded diced lemon, or strips of lemon zest.