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.
Notes
TO STORE: Put in the airtight container to store in the refrigerator for up to a week. TO FREEZE: Place the pesto aioli in a shallow container and store it in the freezer for up to two months.