Throw the basil, olive oil, garlic, and lemon zest into a blender or food processor. Run that shit until the basil is all chopped up and it looks mostly smooth in there. You can totally do this with a mincing with knife if you want but that shit will take a minute.
When the basil mixture is looking right, mix in the mayo until it's all combined. This will keep in the fridge for a week but it will start looking ugly after 3 days.