To a saucepan over medium-high heat, add the water and the sugar. Stir until the mixture heats up and the sugar dissolves, about 2-3 minutes. Remove from heat. Measure 1 cup of the mixture, also known as simple syrup (save any remaining for another use or discard). To the 1 cup of simple syrup whisk in the cocoa powder (you have to do this while the simple syrup is still hot or else the cocoa will clump), followed by the vanilla and peppermint. Allow the mixture to cool 5-10 minutes and then add it to the 2c of half and half. Stir well.
Keep in a sealed container (I use a Mason jar) in the refrigerator. Use as you would store-bought coffee creamer. Shake or stir before each use as the cocoa will settle to the bottom.