Melt the chocolate chips in the microwave on 50% power in 30-second increments, stirring after each, until melted and smooth.
Combine the melted chocolate, sugar, cornstarch, cocoa powder, and milk in a medium saucepan over medium heat and cook, stirring frequently, until the mixture thickens, about 10 minutes. Remove from heat, add the vanilla and butter, and stir until well combined.
Let the mixture cool slightly (it should still be warm but not hot to the touch), and then pour into 8 popsicle molds. Freeze for 30 minutes, and then insert sticks. Freeze completely before serving.