Place the cocoa powder in a large mixing bowl and pour in the boiling water and Kahlua.
Stir with a wooden spoon or rubber spatula until the cocoa comes together into a soft mass.
Add the butter and blend with an electric mixer on low speed until the mixture is soft and well combined, 30 seconds. Stop the machine. Add the confectioners' sugar and cinnamon and beat with the mixer on low speed until the sugar is incorporated, 1 minute. Increase the mixer speed to medium and beat until the frosting lightens and is fluffy, 2 minutes more.