In a large measuring cup or bowl, whisk together the half and half, condensed milk and vanilla bean paste. Fill up the popsicle molds with about 1 to 2 ounces of cream mixture. Place the pops in the freezer to let this layer firm up for 30 to 60 minutes.
Heat the sugar, cornstarch, cocoa, coffee and milk in a saucepan over medium heat, stirring constantly until until the mixture thickens. Remove it from the heat and whisk in the vanilla extract and the butter. Let the mixture cool to room temperature.
Once cooled, remove the molds from the freezer and finish filling each mold with the cold brew mixture. Freeze the pops for 4 to 6 hours.