Place the half, and half heavy cream and mint leaves into a saucepan and heat the cream mixture for 4-5 minutes until bubbles appear on the surface of the cream.
Remove the mint-infused cream from the stove and allow it to steep for 30 minutes to an hour.
Whisk the eggs and granulated sugar in a large bowl until they are pale yellow in color.
Remove the mint leaves from the cream and return it to the stove for 2-3 minutes just before it starts to boil.
Add the unsweetened cocoa powder to the egg yolks and whisk to combine.
Add a ladle full of the mint-infused cream to the egg cocoa mixture and whisk to combine.
Gradually whisk in the remaining mint-infused cream, then place it back into the saucepot and heat it for 5-8 minutes over medium-low heat until it thickens.
Add the bittersweet chocolate and stir until it melts.
Remove the chocolate mint ice cream base from the stove, pour it into a fine-mesh strainer set over a bowl, and allow the chocolate custard to cool completely.
Stir in the vanilla bean paste, then cover the chocolate mint ice cream base press a sheet of cling wrap directly onto the chocolate mint custard and refrigerate it for at least 6 hours overnight.
Churn the chocolate mint ice cream in an ice cream machine, according to the manufacturer's instructions.
Transfer the chocolate mint ice cream to a freezer-safe container and freeze it for 4-6 hours before scooping.