Make the Cupcakes: Preheat oven to 350 degrees F. Line a standard-size muffin pan with liners. Place the chopped chocolate and cocoa powder in a medium bowl. Pour the hot coffee over the mixture and whisk until smooth. Refrigerate mixture for 20 minutes.
Meanwhile, whisk together the flour, sugar, salt and baking soda in a medium bowl; set aside.
Whisk the oil, eggs, vinegar and vanilla extract into the cooled chocolate mixture until smooth. Add the flour mixture and whisk until smooth.
Divide the batter evenly between the muffin pan cups. Bake until the cupcakes are set and just firm to the touch, 17 to 19 minutes. Cool the cupcakes in the pan on a wire rack for 10 minutes, then remove the cupcakes from the pan and place on the wire rack to cool completely, about 1 hour.
Make the Frosting: Using the wire whisk attachment of your stand mixer, whip the butter on medium-high speed for 5 minutes, stopping to scrape the bowl once or twice. Reduce the speed to low and gradually add the powdered sugar. Once all of the powdered sugar is incorporated, increase the speed to medium-high and add the vanilla, mixing until incorporated. Whip at medium-high speed until light and fluffy, about 2 minutes, scraping the bowl as needed. Add the heavy cream a little at a time, whipping after added, until desired consistency is reached.
Frost the cupcakes as desired (I used the Wilton 1M decorating tip). Frosted cupcakes can be stored in an airtight container at room temperature for up to 3 days.
Notes
You can substitute all-purpose flour for the bread flour if needed, but bread flour is recommended.