Step 1Weigh the fish. Measure the salt: For every pound of fish allow a scant 1 tablespoon coarse salt or 2 teaspoons fine salt. Rub the fish with the salt on both sides and in the cavity and set aside on a plate for 30 to 60 minutes.
Step 2To prepare basil oil, blanch the leaves in rapidly boiling salted water just until they wilt, about 15 seconds. Remove immediately and place in an ice water bath to stop cooking. Squeeze the basil dry and chop coarsely.
Step 3Place the basil in a blender and add just enough oil to cover. Puree until finely chopped and then with the motor running, add the remainder of the olive oil through the feed tube on the top. Add salt to taste, about one-fourth teaspoon.
Step 4When the basil mixture is perfectly smooth, pass it through a fine mesh strainer into a measuring cup or bowl. You can stir the mixture gently with a rubber spatula to make it flow a little faster, but be careful not to press -- that will cloud the oil. You will have 3/4 cup to 1 cup of basil oil, which will keep in the refrigerator, tightly covered, for about 1 week.
Step 5Heat the grill or broiler. If using a broiler, line the broiler pan with foil and brush with 1 to 2 tablespoons olive oil. Rinse fish under running water and pat dry with paper towels. Rub lightly with 1 tablespoon olive oil and grill over moderately high heat or broil about 5 inches from the flame until a knife penetrates the flesh and the top fillet begins to lift easily, about 5 to 7 minutes, depending on size of fish. Turn and continue cooking until done, about 5 to 7 minutes more.
Step 6Remove the fish to a platter and let it stand briefly. Lift off the fillets and drizzle them each with about 1 tablespoon of basil oil.