Separate the egg yolks from the egg whites. The whites aren't required for this dish, so you can use them for another dish or freeze for later use.
If making the mayonnaise by hand, place the egg yolks in a large, clean bowl. Place a damp cloth under the bowl to secure it.
Add the mustard and vinegar. Whisk with a balloon whisk to blend. Add a very small amount of the oil and whisk until it's well blended in. Add a little more oil and whisk again.
Continue adding the oil, whisking thoroughly between each addition, until the sauce emulsifies and thickens after a few minutes. Stop adding the oil when the mayonnaise has reached the desired consistency (you may not need to use all the oil). If the mayonnaise is too thick, you can thin it with a little bit of warm water.
Adjust the flavouring by adding more vinegar or mustard, if desired, and season with sea salt and white pepper.
If using a food processor, place the egg yolks into the bowl of a processor, or in a mini-food processor. Add the vinegar, salt and pepper and the mustard. Pulse once or twice to blend.
Turn the processor on and, with the motor running, add the oil in a thin, steady stream through the top of the lid. After a minute or two, the mixture will change consistency as you blend and will emulsify into a thick, rich sauce. Stop adding oil when the mayonnaise has reached the desired consistency (you may not need to use all the oil). If the mayonnaise is too thick, you can thin with a little bit of water.
Taste, adjust the flavouring by adding more vinegar or mustard, if desired, and season with a little more sea salt and white pepper to taste.
The mayonnaise can be served at once or kept in the fridge for up to a day.