Pat fish dry on paper towels. Place oil in a pan just large enough to hold the fish, but with enough room so that the pieces don’t touch each other. Place on the stove and warm oil until tepid. Add basil leaves and try to make them settle on the bottom of the pan. Sit fish on top and turn the heat up. You may need to add a little more oil so that it covers the fish. Bring to a very gentle simmer and cook until fish is barely done, about 8-12 minutes, depending on the thickness of the fish.
To test, carefully lift a piece out with a slotted spoon and pull back the flesh with a sharp knife – it should be translucent. You can serve it now, or leave it to cool completely in the oil. Remove the fish with a slotted spoon and serve with the salad.
If making this in advance, store the fish in the oil, covered tightly in the fridge. Bring back to room temperature before serving.