You can make this sauce up to several days ahead to serve with peeled, boiled shrimp. Leftover sauce will keep in the refrigerator for 2 weeks and is delicious on sandwiches, chicken or crab cakes, or salad.
Process first 12 ingredients in a blender or food processor until smooth. With blender running, pour oil in a slow, steady stream until thickened. Stir in green onions, salt, and black pepper; cover and chill until ready to serve.