Combine cocoa and sugar in a medium saucepan. Whisk in 3/4 cup of the water, a little at a time. Bring the mixture to a simmer. Simmer for 2 minutes, stirring constantly. Remove from heat and stir espresso powder and milk. Strain into a shallow pan. Freeze several hours until hard.
Break the frozen mixture up with a fork and place the chunks in the bowl of a food processor. Add the remaining 1/4 cup of water and process until no lumps remain and the mixture is thick and slushy and lightened in color. Serve at once in goblets to be eaten with a spoon, or refreeze overnight and scoop like sherbet to serve.