In a large saute pan, heat 3/4 cup of olive oil over a medium flame. Add the anchovies and garlic to the hot oil. Using a wooden spoon, press the anchovies to break them up. They will appear to dissolve in the oil. Toss in the eggplant and cook until golden on all sides, about 3 minutes per side. Remove the eggplant from the oil and drain on paper towels.
Pour the excess oil from the pan and discard. Add 1/2 cup of the remaining olive oil to the pan and heat over a medium flame. Toss in the onion and chili flakes and saute until the onion is translucent, about 3 minutes. Hand-crush the tomatoes into the pan and stir well. Bring to a simmer for 10 minutes and then return the eggplant to the pan.
Add the black olives, capers, raisins, and pine nuts to the pan, stir well, and sprinkle with the sugar and vinegar. Pour in the tomato sauce and stir; taste and then season with sea salt and freshly ground black pepper. Cover and simmer until ready to serve.
Meanwhile, coat the swordfish fillets in the remaining 1/4 cup olive oil, lemon juice, and basil. Marinate in the refrigerator for 20 minutes.
Heat a grill pan over a medium flame, and brush with oil to prevent the fish from sticking. Season the fish with sea salt and freshly ground black pepper and then place in the pan. Sear on both sides for 3 to 5 minutes per side, or until fully cooked through (opaque in the center). Serve immediately with a generous spoonful of the caponata.