To make the mayonnaise, in a food processor or blender, combine the egg, lemon juice and Dijon mustard. Process on high speed for 15 seconds. With the motor running, pour the oil through the feed tube in a slow, steady stream and process until emulsified. Add the salt and cayenne, and pulse to blend.
To make the tartar sauce, transfer to a medium bowl and fold in the remaining ingredients. Cover and refrigerate for 1 to 2 hours before serving. (NOTE: Both mayonnaise and tartar sauce should be used within 24 hours.)
Yield: about 1 1/4 cups