Make the Ganache Filling: Place the chopped chocolate, heavy cream and powdered sugar in a small bowl. Heat in the microwave on high power until the mixture is warm to the touch, 20 to 30 seconds. Whisk until smooth, then refrigerate until just barely chilled, no longer than 30 minutes.
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. Place one slightly rounded teaspoon of ganache filling on top of each cupcake. 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: In a food processor, process the butter, sugar, cocoa powder and salt until smooth, about 30 seconds, scraping the sides of the bowl as needed. Add the corn syrup and vanilla extract and process until just combined, 5 to 10 seconds. Scrape the sides of the bowl, then add the melted chocolate and pulse until smooth and creamy, 10 to 15 one-second pulses. Frost the cupcakes as desired. The frosted cupcakes can be kept in an airtight container at room temperature for up to 3 days.
The ganache filling can be omitted it if you'd like a more traditional chocolate cupcake.
The frosting can be kept at room temperature for up to 3 hours before using, or refrigerated for up to 3 days. If refrigerated, let stand at room temperature for 1 hour before using.