Bring a large pot of water to a boil. Fill a bowl halfway with ice water.
Set broiler on high. Place the salmon on a foil-lined broiler pan and season with 1/4 teaspoon of the salt. Cook, without turning, until the salmon is the same color throughout and flakes easily, 7 to 10 minutes, depending on the thickness; set aside.
Meanwhile, add the basil to the pot and cook for 5 seconds. Using a slotted spoon, transfer the basil to the ice water. Drain and squeeze to remove any excess water. Place on a paper towel and pat dry. Place the basil, oil, pepper, and the remaining salt in a blender and puree until smooth. Transfer the salmon to individual plates and drizzle with some of the basil oil. (You can cover and refrigerate the remaining oil for up to 2 weeks.)
Tip: You can also drizzle this aromatic oil on corn on the cob, grilled or broiled chicken, fresh mozzarella or Feta, boiled new potatoes, roasted vegetables, fried eggs, country-style bread, and vegetable or white bean soup.