Preheat the oven to 350 degrees F. Spray two 8-9 inch round cake pans with nonstick baking spray. Cut parchment paper circles to fit in each pan and press them again the bottoms. In the bowl of an electric mixer combine the sugar, flour, cocoa powder, baking powder, baking soda, and salt. Stir to blend. Add the sour cream, eggs, oil and vanilla. Beat to combine. Then scrape the bowl, turn the mixer on low, and slowly add in the hot coffee. Beat until the batter is thin and smooth.
Pour the batter evenly into the two prepared cake pans. Bake for 30-35 minutes, until a toothpick inserted into the center of the cake comes out clean. Cool the cakes in the pans for 5 minutes. Then flip the cakes out onto cooling racks, remove the parchment paper, and cool completely.
Meanwhile, line an 8- to 9-inch cake pan with plastic wrap. Scoop the ice cream into the cake pan and press down with a spatula to make a smooth solid ice cream layer. Place the ice cream layer back into the freezer to firm up.
Once the cakes are cool, wrap each layer individually in plastic and freeze them until they are cold and hard to the touch, around 2 hours.
For the ganache: Heat the cream until it nearly simmers. Then remove from heat and add broken chocolate pieces to the cream. Allow the mixture to sit for 10 minutes. Then add the corn syrup and gently stir until completely smooth. (Whisking adds bubbles, so stir gently instead.) Allow the ganache to cool to room temperature before proceeding.
To assemble: Unwrap the frozen cakes. Lay one cake on a cake stand then flip the ice cream layer on top of it and peel off the plastic. Top with the second cake layer and press down a little to level the cake. Use a flat spatula to smooth out the seams between the ice cream and cake all the way around the edges. Then pour the ganache over the top. Spread to the edges, if needed, to help the ganache drip over the sides. Freeze until ready to serve.
NOTES: This cake can be made with any variety of ice cream or gelato.You can skip the corn syrup in the ganache, but it will not look as glossy.Allow the cake to rest on the counter for about 10 minutes before cutting.