Warm a pan over a medium heat and add the pine nuts. Toast for a couple of minutes until golden. Keep an eye on them and keep shaking/stirring as they can easily burn.
Remove the pine nuts from the pan and allow to cool in the mortar or blender bowl.
Add the garlic and a pinch of salt and pepper. Pound with the pestle (or pulse in the blender) until coarsely crushed.
Add the basil. If you're using a pestle and mortar then add a little at a time. If you're using a blender, you can add all at once. Pound with the pestle or pulse with the blender to a coarse mixture.
Add the cheese and olive oil. Pound/pulse again until you have a pesto. I like to keep it coarse, but you can break it down as much as you like.
Cover and set aside while you make the mayo.
Make the mayo
Put the vinegar, egg yolks and mustard in a large, clean bowl. Beat with an electric mixer until well combined.
Add 1 tablespoon of the olive oil. Beat until it emulsifies – it will look thicker, paler and creamier.
Keep adding the oil a little at a time, while beating at top speed all the time. When you have run out of olive oil, continue with the vegetable oil.
Keep beating until all the oil is incorporated. You should have a pale, creamy mayo that can just hold its shape. See notes on thickness.
When you've reached your preferred mayo thickness, transfer it to a serving bowl and spoon the pesto on top, swirling to semi-combine. Serve.
I think the thickness of this mayo is perfect for swirling with the pesto, but if you prefer a thicker, more scoopable mayo (like the popular store bought brands), you can whisk in up to an extra 100ml (3.25 floz) of vegetable oil. Don't add it all at once, since you can always add more, but you can't take away.Conversely, if the mayonnaise feels too thick once all of the oil is added, you may loosen by beating in 1-2 tsp of vinegar. Again, add slowly so that you don't overdo it.The nutritional estimates are based on approximately 1 tbsp of mayo and do not include any extra oil or vinegar.This makes a big batch of mayo - if you want a smaller amount, just head up to the slider and change the number of servings to 16, then you'll get all the right quantities to make this with one egg.