Mix all of your ingredients together into a sauce pan and whisk well.
Stir over low-medium heat for 10-15 minutes to get a slow reduction, heat the egg yolk and melt the sugars down. Be sure not to overheat or else you'll curdle your milks and cook your eggs. Stir consistently.
Strain your mixture into a large jar and then add more fresh basil leaves. Let sit six hours - overnight in the fridge for the basil flavor to infuse.
When ready to make your ice cream, strain basil leaves out again. Pour mixture into ice cream maker.
Turn your ice cream maker on and let it churn based on manufacture's recommendations. Usually 20 - 30 minutes. Ice cream should double in volume and be solid, but soft when done.
Notes
Add mixture to an air tight, sealable container and freeze for at least three hours before serving.