Place everything but the olive oil in a food processor or blender. Pulse a few times to combine, using a spatula to scrape the sides and push everything together.
In a blender, remove the cap on the lid. Slowly begin to drizzle the oil in the machine. If using a food processor, slowly pour the oil in while the machine is running. If using a blender, hold your hand over the lid as it will splatter if you don't.
When oil is incorporated and the basil and shallot are pureed, turn off and scrape the sides again. Cover and blend or puree for 1-2 more minutes.
Place in a glass jar, cover and store in the refrigerator for up to two weeks. When using the leftover dressing, let it sit at room temperature about five minutes before serving. Shake well and use liberally.