Add the icing sugar, and unsalted butter to a bowl of a stand mixer fitted with a paddle attachment. Beat at medium speed until combined, the mixture should look like fine breadcrumbs
Add the vanilla to the milk and slowly pour this into the powdered sugar mixture with the mixer running at low speed.
Once all of the milk has been added scrape down the sides of the bowl. Mix the frosting again at medium-high speed for about 5 minutes or until the frosting is light and fluffy. If the frosting seems too thick beat again.
Scrape down the sides of the bowl often with a rubber spatula when beating to ensure that the frosting is beating thoroughly.
In a small bowl combine the hot water and coffee, and allow the coffee to dissolve. Add the coffee mixture to the buttercream and beat again at medium-high speed. The coffee buttercream is ready to use on cakes and cupcakes.