Melt the chocolate chips in a microwave at 50% power in 30-second increments until the chocolate is melted and smooth when stirred.
Heat the sugar, cornstarch, cocoa, melted chocolate, coffee, and milk in a saucepan over medium heat, stirring constantly until the mixture thickens. Remove from the heat and whisk in the vanilla extract and the butter.
Let the cold brew chocolate mixture cool slightly (close to room temperature and not hot to the touch); pour into 8 popsicle molds. Freeze for 30 minutes, and then insert sticks. Freeze for an additional 4 to 6 hours.