In a blender or food processor, blend together the egg, lemon juice, mustard, fine sea salt, and freshly ground white or black pepper, blending until well combined. With the motor still running, add the oil in a very slow, thin, steady stream and blend until the dressing is thick and smooth. DO AHEAD: The mayonnaise can be prepared ahead and refrigerated, in an airtight container, up to 1 week.
