In a small saucepan, combine cocoa and sugar. Stir in just enough of milk to form a smooth paste. Stir in all but 2 tablespoons of remaining milk, stirring over low heat until sugar is completely dissolved.
Pour into ice-cube trays, cover well. Freeze at least 6 hours or overnight (or up to 1 week or more).
Place cubes in food processor bowl fitted with steel blade. Process with remaining 2 tablespoons milk until no lumps remain and mixture is lightened in color. Serve in frosted goblets as a spoon drink or cover, and refreeze at least 8 hours. Scoop, serve frozen.