In a medium saucepan, whisk together the sugar and flour. , Add the milk and cook over medium heat, whisking constantly, until the mixture comes to a boil and has thickened to a pudding-like consistency, 3 to 6 minutes. , Remove the mixture from the heat and transfer it to the bowl of a stand mixer fitted with the flat beater attachment. Beat at medium-high speed for 10 to 15 minutes, until it's below 80°F. (Measure the frosting’s temperature with a digital thermometer, if possible. If you don’t have a thermometer, it should feel slightly cooler than body temperature.)  , Meanwhile, melt the chocolate and espresso powder in a double boiler or in short intervals in the microwave until smooth. Set aside.  , Reduce the mixer speed to low and, with the mixer running, add the butter a few tablespoons at a time. Scrape down the sides of the bowl as needed, to make sure the butter is thoroughly incorporated before adding more.  , Once all the butter has been mixed in, add the vanilla.  , Increase the mixer speed to high. Beat the frosting for 4 to 6 minutes, until it’s light and fluffy., Add the melted chocolate and beat until smooth. , If the frosting feels dense, curdled, or greasy, it’s too cold. Gently warm the mixture by setting the bowl over a pan of steaming-hot water for a few minutes (just until you see the edges begin to soften/melt), then re-whip until smooth. If the frosting feels soupy or loose, it’s too warm. Chill for 15 to 20 minutes in the refrigerator before re-whipping. , Use the chocolate ermine frosting as desired to finish the cake or cupcakes of your choice; you’ll have enough for a single-layer cake, a 6" two- or four-layer cake, or a dozen cupcakes (see “tips,” below). , Store the chocolate ermine frosting, in an airtight container, at room temperature for up to 2 days or in the refrigerator for up to 1 week; freeze for longer storage. If stored in the refrigerator, let it sit at room temperature for 30 minutes, then gently beat it to return it to the proper consistency before using.