Set a fine mesh strainer over a large bowl, resting in a ice bath. Place chocolate in a medium heatproof bowl.
In a medium, heavy-bottomed pot, whisk together egg yolks, sugar, and cocoa powder until well combined. Whisk in cream and milk until yolk mixture is fully incorporated. Cook over medium heat, whisking frequently, until custard thickens enough that a finger swiped across the back of a wooden spoon leaves a clean line and the temperature reaches 170°F on an instant-read thermometer.
Pour mixture over chopped chocolate. Allow mixture to rest for 2 minutes until chocolate is melted, whisk until chocolate is completely incorporated into custard base.
Immediately pour chocolate custard through sieve into prepared ice bath. Stir salt and vanilla into the custard. Occasionally stir custard while it is in the ice bath. Once custard is at room temperature, transfer to refrigerator to chill fully.
Churn custard in ice cream maker, until the texture of soft serve, according to manufacturer's instructions. Transfer to an airtight container and freeze until firm enough to scoop, about two hours.