Preheat oven to 350°F (177°C). Grease two 8-inch (20 cm) round cake pans with butter, line their bases with parchment paper, then grease and dust with cocoa powder. This ensures easy release and crisp sides.
In a large bowl, sift together the flour, Dutch-process cocoa powder, natural cocoa powder, baking powder, baking soda, and salt. Whisk thoroughly, reaching across the bottom and around the sides of the bowl, until the mixture is evenly colored and no cocoa pockets or visible clumps remain.
In a stand mixer bowl, beat eggs with sugar until pale and slightly thickened, about 2-3 minutes at medium speed. Reduce to low, then drizzle in oil and beat until smooth and emulsified.
Add vanilla and peppermint extract to the mixture. Alternately add dry mix and buttermilk, beginning and ending with dry ingredients. Scrape sides and mix just until the last streak disappears, overmixing toughens crumb.
With the mixer running on low speed, slowly pour the hot water into the batter so it blends in without splashing. Stop once midway to scrape the sides and bottom of the bowl, then continue mixing just until the batter is smooth, fluid, and evenly chocolate-colored. The finished batter will be thin.
Divide the batter evenly between the two prepared pans, then smooth the tops and tap the pans gently to release large air bubbles. Bake on the center rack for 26-32 minutes. The layers are ready when the centers look set and a toothpick inserted there comes out with moist crumbs but no wet batter.
Let cakes cool in the pans 12 minutes. Run a knife along edges, then invert cakes onto a rack. Peel off parchment and cool completely; this prevents steamy, soggy layers.
Combine egg whites and sugar in a heatproof bowl. Set over simmering water; whisk gently and heat to 160°F (71°C), until sugar dissolves. Transfer to a mixer, whip to stiff peaks and cool completely (12-15 min).
Switch to the paddle attachment. On medium-low, add cubed butter a few pieces at a time, beating fully between each. Once thick and creamy, beat in peppermint and vanilla extracts until smooth and fluffy.
Level cake layers. Place one layer on a stand, spread about 1 cup frosting evenly, edge to edge. Top with second layer. Apply a thin crumb coat, chill 15 minutes, then cover with remaining buttercream, smoothing tops and sides.
For ganache, gently heat cream until steaming, pour over chopped chocolate, and sit 3 minutes. Stir until glossy and smooth. Cool until thick but pourable, then drip over chilled frosted cake with a spoon or piping bag.
While the ganache is still tacky, sprinkle the crushed candy canes over the top so they adhere without sinking. Chill the finished cake for 20-30 minutes before slicing. The buttercream and ganache should feel set, and a knife should pass cleanly through the layers for neat, defined servings.
