Ensure everything is at room temperature before you get going. This makes the finished product much less likely to split.
Put the egg yolk and salt in the bowl. Whisk for about 20 seconds, until slightly thickened, then add the mustard and vinegar. Whisk for another 30 seconds.
Begin to add the neutral oil into the bowl, drop by drop, whisking all the time.
Keep whisking and pouring until the mayonnaise begins to thicken, at which point you can begin to add the oil a little more quickly, followed by the olive oil. Once both oils have been incorporated and you have a thick, glossy mixture, season to taste. If you prefer a thinner consistency, beat in a little water, lemon juice or vinegar.
If your mayonnaise splits, it’s easily rescued. Put another egg yolk into a fresh bowl and whisk for a minute until thick and sticky, then gradually beat into the split mayonnaise a little at a time, followed by any remaining oil.