How to Make It
Peel the eggplant and cut into 1-inch pieces. Heat a large heavy-bottomed saucepan with half of the extra virgin olive oil; add half of the eggplant and season with 1/4 teaspoon of kosher salt. Fry the eggplant cubes in two batches until golden brown, adding the second addition of olive oil and salt with the second batch. Remove from the oil and set aside.
Using the same pan, add the celery, red onion, garlic and capers and season with salt and pepper. Cook over medium to low heat until vegetables have softened and look glossy, about 10 minutes. Add the diced tomatoes and sugar to the pan and stir to combine. Fold in the cooked eggplant and cook over medium heat for about 20 minutes, stirring every 4 minutes. Add the vinegar to the mixture and season with salt and pepper to taste.
Season the halibut with salt and pepper, add to a large skillet with 3 tablespoons of extra virgin olive oil and cook over moderately high heat until browned, about 4 minutes. With a fish spatula, flip each filet in the pan, and continue cooking for another 3 minutes. Transfer to a plate and serve with the caponata. Garnish with fresh basil and serve.