Drain cod and cut into 3-inch pieces. Transfer to a large saucepan, cover with COOL water, and bring to a boil over medium-high heat. Reduce heat to medium and cook until fish is easily flaked with a fork, 15 to 20 minutes. [Cod will still be firm after cooking.]
Meanwhile, in a large bowl, whisk together olive oil, lemon juice, olives, garlic, and basil; season with salt and pepper.
For cold salad: Drain cod and transfer to a bowl filled with cold water until cooled, 10 to 15 minutes.
For warm dish: pat cod dry with paper towel and remove any bones.
Drain cod and flake with a fork or fingers. Add to bowl with olive oil mixture; toss to coat. May serve warm or room temperature. For cold salad: Refrigerate at least 1 hour before serving. Serve chilled or at room temperature.
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