Combine the heavy cream and milk in a medium saucepan. Heat over medium-low heat until just barely simmering. Add the basil and mint and remove from the heat. Let the mixture cool for 1 hour.

In a medium bowl, whisk the sugar, egg yolks, and salt.

Strain the cream mixture through a fine mesh sieve, discarding solids. Return to the saucepan and heat over medium-low heat until just barely simmering.

Carefully ladle 1/2 cup of the hot cream into the bowl with egg yolk mixture and whisk to combine. Then, whisk the egg yolk mixture back into the saucepan with the cream.

Cook, stirring constantly, until the mixture thickens and coats the back of a wooden spoon (it will register between 165° and 170° on a thermometer). Remove from the heat and stir in the vanilla.

Transfer the mixture to an airtight container and chill until very cold, at least 4 hours and preferably overnight. Freeze in an ice cream maker according to the manufacturer's instructions. In the last few minutes of churning, add the chocolate chips.