Fill a bain-marie pan (see Notes) with 1 inch of water. Place the chocolate and coffee in a heatproof bowl or pan, and place over the pan of water. Heat, stirring occasionally. If the heat comes to a simmer, turn it off and let the chocolate continue to melt until velvety smooth.
Remove the bowl of melted chocolate from the hot water bath and with an electric mixer, whisk or wooden spoon, beat the butter into the chocolate, around one tablespoon at a time.
Submerge the bottom quarter of the bowl in ice water and beat the frosting at medium-high speed until of good spreading consistency. (After about 5 minutes it will lighten in color, signaling that it will be ready quickly.)
Use to frost immediately. Serve at room temperature.
Notes
Melting the Chocolate - If you don’t have a bain-marie set (double boiler), you can use a tall-sided heatproof bowl or pan set over a second pan.
Storing - The frosting can be stored at room temperature, covered and kept away from sunlight and heat, for a few days, kept in an airtight container in the refrigerator for up to a week, and in the freezer for up to a month. Let come to room temperature before using. Whip, whisk or stir to restore a smooth, fluffy texture.