Put the chocolate in a microwave-safe bowl and microwave in 40-second intervals, stirring after each, until just melted. Whisk until smooth; set aside.
Combine the milk, sugar and egg yolks in a medium saucepan. Cook over medium heat, whisking, until the sugar dissolves and the mixture is thick enough to coat a spoon, about 7 minutes. Strain the custard through a fine-mesh sieve into a blender. Add the melted chocolate and pulse until smooth, about 1 minute. Add the heavy cream and vanilla and pulse until just combined, about 30 seconds. Transfer to a bowl, cover with plastic wrap and refrigerate at least 3 hours or overnight.
Transfer the chilled custard to an ice cream maker and churn according to the manufacturer's instructions. Transfer to a resealable container and freeze until firm, about 2 hours.