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 ¼ teaspoon salt.
Cook the salmon, without turning, until it is the same color throughout and flakes easily, 7 to 10 minutes, depending on the thickness. Transfer to plates.
Meanwhile, add the basil to the pot of water and cook for 5 seconds. Using a slotted spoon, transfer it to the ice water. Drain, squeeze to remove excess water, place on a paper towel, and pat dry.
Place the basil, oil, ¾ teaspoon salt, and ¼ teaspoon pepper in a blender and puree until smooth.
Drizzle the salmon with some of the basil oil and serve. (The remaining oil can be refrigerated, covered, for up to 2 weeks.)