Put the ice cream container from your ice cream maker in the freezer.
Heat the milk and cream with the zest and basil along with a pinch of salt. Bring it up to a gentle boil, then shut of the heat and let it infuse. Make sure to stir so the bottom doesn't scorch. Let sit and infuse until no longer hot, but still warm.
Separately, whisk together the egg yolks and the sugar until they are light in color from the air that you've incorporated. Set aside.
Mix the basil milk in a blender until uniform.
Temper the yolks by adding a bit of the warm liquid to them gradually. Keep stirring constantly, scraping down the sides to avoid scrambled eggs.
Return it all to the sauce pan and stir it constantly over low heat until it thickens to the point where you can leave a track with your finger on the spatula.
Let it cool for 10-15 minutes, then pour the liquid into the frozen ice cream container and run it through the ice cream maker for 50-60 minutes.