Line an 8-inch round cake pan with parchment on the bottom and lightly grease the sides. Place the bowl and whisk attachment for your mixer in the refrigerator to chill. Set the ice cream in the refrigerator to soften while you bake (20–25 minutes does the trick). Preheat oven to 325°F (165°C).
Bloom the cocoa: In a heatproof bowl, whisk the cocoa powder and espresso powder with the hot coffee until smooth and glossy. Let cool 5 minutes.
In a separate bowl, whisk together flour, baking powder, baking soda, and salt until evenly combined.
In a large mixing bowl, whisk the sugar, eggs, vanilla, and sour cream until the mixture looks thick and slightly pale, 45–60 seconds. Stream in the melted, cooled butter and whisk until fully incorporated.
Add about half the dry ingredients to the wet mixture and whisk just to combine. Pour in all of the cocoa-coffee mixture and whisk until smooth, then fold in the remaining dry ingredients with a spatula until no dry streaks remain. The batter will be fluid.
Bake: Transfer batter to the prepared pan and bake at 325°F for 32–35 minutes, until a toothpick comes out with a few moist crumbs. Cool 10 minutes in the pan, then turn out onto a rack to cool completely.
Level and make crumbs: Slice off any domed top to create a flat cake. Crumble the trimmings into small bits. For extra texture, spread crumbs on a sheet pan and toast in the warm oven for 5 minutes to dry; cool completely.
Set the cooled cake into an 8-inch springform pan (or place a parchment sling in the same cake pan to help lift later). Spread half of the softened ice cream over the cake in an even layer. Sprinkle most of the cake crumbs over the ice cream, gently pressing them in, then top with the remaining ice cream and smooth the surface. Freeze until solid, 3–4 hours.
Make the chocolate whipped cream: In the chilled mixer bowl, beat the cold heavy cream on medium-high to soft peaks. With the mixer running, slowly add the sifted powdered sugar and cocoa, then mix in vanilla. Continue whipping to medium-firm peaks; do not overbeat.
Unmold and finish: Release the springform ring (or lift out using the parchment). Frost the top and sides with the chocolate whipped cream, or pipe decorative swirls. Garnish with dark chocolate shavings and any remaining cake crumbs. Freeze 20–30 minutes to set before slicing. Tip: Warm a sharp knife under hot water and wipe dry between cuts for extra-clean slices.