In a medium saucepan, combine heavy cream and granulated sugar over medium heat. Stir constantly until the sugar fully dissolves, then remove from heat. Allow mixture to cool slightly before whisking until smooth. Transfer to the refrigerator to set for about 30 minutes.

In a large mixing bowl, sift together all-purpose flour, unsweetened cocoa powder, baking soda, and baking powder. Whisk in dark brown sugar to form the dry mixture.

In a separate bowl, beat together eggs and vegetable oil until well combined. Add hot coffee and instant coffee, mixing thoroughly. Gradually combine with dry ingredients, stirring until just mixed.

Preheat the oven to 350°F (175°C) and grease two 9-inch round cake pans. Divide the batter between the pans and smooth the tops. Bake for 30-35 minutes, or until a toothpick comes out clean.

Once baked, let the cake layers cool in the pans for about 10 minutes. Carefully transfer them to wire racks to cool completely.

Mix brewed coffee with a tablespoon of sugar until dissolved. Brush this coffee soak generously over each layer once cooled.

Level the tops of each cake layer if needed. Spread frosting between the layers and apply a thin crumb coat over the entire cake. Chill for 15 minutes.

Generously frost the top and sides of the cake with the remaining frosting, smoothing the finish.