Put the green onions, garlic, horseradish, mustards, ketchup, hot sauce, Worcestershire, lemon juice, egg, salt, and cayenne in a food processor or blender and process until smooth about 30 seconds. With the motor running, slowly pour in the vegetable oil through the feed tube. The mixture will thicken. Transfer to a container, cover, and refrigerate for 1 hour before using. Best if used within 24 hours.