Position a rack in the center of the oven and preheat to 350°F (177°C). Lightly grease three 8-inch round cake pans, line each base with parchment, then grease the parchment. Dust the sides with cocoa powder and tap out the excess. Set the pans on a digital scale one at a time and note their empty weights if you want to divide the batter with maximum consistency.

Place a fine-mesh sieve over a large mixing bowl and sift in the flour and Dutch-process cocoa. Add the sugar, baking soda, baking powder, and salt, then whisk for about 45 seconds. The mixture should look evenly colored with no cocoa pockets or compacted leavener. Thorough distribution matters here because this thin batter is mixed only briefly after the flour becomes wet.

In a separate medium bowl, whisk the room-temperature eggs until the whites and yolks are fully blended. Add the buttermilk, sour cream, oil, and vanilla, then whisk until smooth with no visible streaks of sour cream. Pour the wet mixture into the dry ingredients and mix on low speed for 30 seconds. Stop, scrape the bowl and paddle, then mix on medium-low just until the batter is uniform, about 45 seconds.

With the mixer running on its lowest speed, pour in the hot coffee gradually so it incorporates without splashing or shocking the eggs. Stop once the batter is glossy, fluid, and evenly chocolate-colored, about 30 seconds, then scrape the bottom with a flexible spatula. Divide approximately 740 grams of batter among each prepared pan. Gently rotate the pans to level the surface, but do not bang them hard enough to remove the air needed for lift.

Bake all three pans on the center rack for 30 to 35 minutes, rotating their positions after 22 minutes if the oven heats unevenly. The layers are done when their centers spring back lightly, a tester shows moist crumbs rather than wet batter, and an instant-read thermometer inserted away from the pan registers about 205°F (96°C). Cool in the pans for 15 minutes, loosen the edges, invert onto racks, remove the parchment, and cool completely for at least 90 minutes.

Place the 750 grams chopped chocolate in a tall heatproof bowl. Heat the cream in a small saucepan over medium heat until steaming and bubbling gently around the edge, without a rolling boil. Pour it over the chocolate, cover, and rest for 3 minutes. Stir slowly from the center outward until glossy, then blend in the butter and peppermint extract without whipping in air. Leave at cool room temperature for 60 to 90 minutes, stirring occasionally, until thick and spreadable.

Use a serrated knife to trim only pronounced domes from the completely cool layers. Secure one layer to an 8-inch cake board with a dab of ganache, cut side up. Spread 225 grams of ganache evenly to within 1/4 inch of the edge. Add the second layer, repeat with another 225 grams, and place the final layer cut side down. Check the stack from several angles, adjust while the filling is soft, then refrigerate uncovered for 20 minutes to stabilize it.

Spread a thin crumb coat of ganache over the top and sides with an offset spatula, filling visible gaps, then chill for 15 minutes. Apply a 1/4-inch final coat and smooth it with a bench scraper, warming and drying the scraper between passes if needed. Transfer the remaining ganache to a star-tipped piping bag and pipe 12 to 16 swirls around the top. Shave the 60 grams of chocolate into curls, scatter them over the swirls, and let the cake set for 30 minutes before slicing.