Rinse and drain basil leaves. Pat leaves dry with a towel. In a blender or food processor, combine basil leaves and olive or vegetable oil (see notes). Whirl just until leaves are finely chopped (do not purée).
Pour mixture into a 1- to 1 1/2-quart pan over medium heat. Stir occasionally until oil bubbles around pan sides and reaches 165° on a thermometer, 3 to 4 minutes. Remove from heat and let stand until cool, about 1 hour.
Line a fine wire strainer with two layers of cheesecloth and set over a small bowl. Pour oil mixture into strainer. After oil passes through, gently press basil to release remaining oil. Discard basil. Serve oil or cover airtight and store in the refrigerator up to 3 months. The olive oil may solidify slightly when chilled, but it will quickly liquefy when it comes back to room temperature.
Nutrition analysis per tablespoon.