In a large mixing bowl, whisk together the cocoa powder and brown sugar until no lumps remain.
Gradually whisk in the cold whole milk until the sugar is completely dissolved and the mixture is smooth. Whisk in the heavy cream and vanilla just until combined.
Cover and refrigerate until thoroughly chilled, at least 2 hours or up to overnight.
Stir the chilled mixture, then pour it into a home ice cream maker and churn according to the manufacturer’s directions until thick and softly frozen, usually 15 to 25 minutes.
Transfer the churned ice cream to a freezer-safe container, spreading it into an even layer. Cover and freeze until firm, at least 3 hours.