Bring the milk to a simmer in a medium-sized saucepan.
Stir in the basil and cinnamon stick, cover, and remove from heat.
Steep 15 minutes, then pour the milk through a sieve, pressing down on the solids to extract all the liquid.
Whisk together the egg yolks and sugar in a large mixing bowl.
Whisk the cinnamon-basil flavored milk.
Pour it back into the saucepan and stir constantly with a rubber spatula over medium heat just until the custard thickens enough to coat the back of a spoon (180 degrees) but no further or it may curdle.
Pour the custard back into the mixing bowl and cool, either over ice or in the refrigerator, until the custard is very cold.
Whisk in the heavy cream.
Freeze in an ice cream maker.
Store in a covered container in the freezer until firm enough to scoop.