In a blender, whiz the mustard, salt, pepper and egg yolk until just blended.
Pulse in 4 tablespoons of the oil, a bit at a time. Then add the remainder of the oil in a steady stream while whizzing. Stop when all the oil is in and the sauce is thick and creamy.
Then blend in the lemon juice and vinegar, and pour the mayonnaise into a bowl or container.
Store in refrigerator covered for up to 3 days.
Notes
If you want, you can also make this in the bowl with a whisk following the same directions.