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1
Make artichoke bottoms:
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2
Break off and discard stems of artichokes. Bend back outer leaves until they snap off close to the base and remove several more layers of leaves in the same manner until pale inner leaves are reached. Trim bases, sides, and inner leaves with a very sharp stainless steel knife. Scoop out chokes with a spoon and rub cut surfaces with 1 lemon half. Drop bottoms into a bowl of cold water acidulated with juice of another lemon half.
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3
Fill a saucepan two-thirds fill of cold salted water. Add juice of remaining lemon half and bring liquid to a boil. Add artichokes and boil 15 minutes, or until just tender.
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4
Drain in a colander.
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5
Make fish:
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6
In a colander sprinkle eggplant generously with salt and let stand 1 hour. Rinse eggplant and drain well. In a skillet heat 3 tablespoons olive oil over moderate heat until hot but not smoking and cook eggplant, stirring occasionally and adding a few tablespoons water if eggplant begins to stick, until very tender, about 20 minutes. Season eggplant with salt and pepper and keep warm, covered.
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7
Cut artichoke bottoms into thin slices and arrange in one layer on a tray or in a shallow baking pan. Brush artichokes with 3 tablespoons olive oil and sprinkle with 3 tablespoons chopped herbs; salt and pepper to taste.
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8
In a small saucepan warm tomatoes over moderate heat until just warmed and season with salt and pepper. Keep warm over very low heat.
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9
Season swordfish with salt and pepper. In a heavy 10-inch skillet heat about 2 cups oil (1-inch deep in skillet) over low heat until just warm to the touch (180 F). Poach fish covered in warm oil 3 minutes; turn with tongs. Poach 3 minutes more, until just cooked through. Transfer to a tray and top with dried tomatoes.
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10
Divide artichokes among 4 plates, mound eggplant in center of circles and top with fish. Scatter with olives and remaining tomatoes; spoon reduction around along with last of oil. Sprinkle plate with remaining herbs.