In a medium saucepan over medium-low heat, add the milk and sugar, stirring until the sugar has dissolved; this will take about 5 minutes. Add in the lemon zest, basil leaves and fresh lemon juice. Turn off the heat and let sit for 30 minutes.
Take the basil out, return to medium-low heat once more and whisk in the egg yolks, one at a time making sure the first is well blended before adding in the second. Add in the salt and gently whisk for about 10 to 15 minutes or until the ice cream base is hot to the touch (it should reach 170 F (77 C) to kill any bacteria from the eggs). Remove from the heat and transfer to a large-lidded container. Add in the basil back to the ice cream mixture. Cover and set in the fridge until chilled through, about 2 to 3 hours.
Once cool, add to your ice cream maker* and proceed with manufacturer's instructions. You want to stop the gelato before it gets too firm, for me this was around 25 minutes in my ice cream maker. At this point drizzle in the olive oil while the machine runs. Scoop into a clean, lidded container and freezer for several hours before serving.