Bring a pot of water to the boil. Meanwhile, fill a large bowl with icy cold water (add some ice cubes in there too!).
When your water has come to a boil, blanche Basil leaves for 5 seconds (no longer), and then immediately remove Basil leaves and plunge into your prepared iced water.
Remove basil from iced water and pat dry with a towel or paper towel.
In a food processor, combine Basil, Garlic, Pine Nuts, Miso, Nutritional Yeast and Salt. Process until blended. Set Pesto aside.
In the blending container that comes with your stick blender, combine all ingredients, except the oil. Mix well and then gently pour oil over your mixed ingredients.
Place your stick/immersion blender into the very very bottom of your blending container and keep it there. Turn your stick blender on to its highest setting, and almost instantly you will start to see the mixture at the bottom thicken.
Now, slowly pull the blender up and through the Vegetable Oil until you reach the surface. Your Vegan Mayonnaise will form relatively quickly. Once formed, continue blending for a few seconds to make sure that everything has emulsified.
At this point, you can add your Basil Pesto into the blending container and blend till you achieve a smooth green Vegan Pesto Aioli, or you can do this by hand for a more rustic and textured Aioli. The choice is yours.
Store Vegan Pesto Aioli in an airtight jar or container. It will keep in the fridge for up to 2 weeks.
Enjoy with French Fries, in a Sandwich or top your Burgers off with the Aioli.