For the Peppermint Ice Cream:
In a saucepan over medium-high heat, whisk together the milk, heavy cream, half the sugar and salt. Heat just until scalding, about 200 F.
Whisk together the egg yolks and the remaining sugar in a medium-sized mixing bowl until smooth and slightly lightened in color. Slowly pour the warm milk mixture into the eggs, whisking constantly until blended. Pour the custard back into the saucepan and cook on medium-low heat, stirring constantly with a spoon, until the custard thickens and a finger drawn across the back of a spoon leaves a path, 5 to 7 minutes.
Pour the custard through a fine-mesh strainer sitting over a clean bowl. Place the bowl in a larger one filled halfway with ice and water and let cool, stirring occasionally. Cover and refrigerate until thoroughly chilled, at least 4 hours.
Place the custard in an ice cream maker and freeze according to the manufacturer's instructions. Transfer the ice cream to a storage container and fold in 3/4 cup of the crushed peppermint candies until thoroughly combined. Cover and freeze until firm, at least 4 hours.
Devil's Food Cake:
Have all the ingredients at room temperature.
Position a rack in the bottom third of the oven and preheat to 350 F. Butter three 9-inch round cake pans and line the bottom of each pan with a round of parchment paper.
Place the chocolate and cocoa in a mixing bowl and whisk in the boiling water until smooth and the chocolate has melted. In another small bowl, whisk together the flour, baking soda and salt. Set aside.
In the bowl of an electric mixer fitted with the flat beater, beat the butter on medium speed until smooth and creamy, 1 to 2 minutes. Add the sugar and continue to beat until the mixture is light and fluffy, stopping to occasionally scrape down the sides of the bowl, about 5 minutes. Stream in the eggs a little at a time, beating until incorporated before adding more. Add the vanilla and beat until incorporated, 30 seconds. Add the buttermilk and beat until combined, about 1 minute. Scrape down the sides of the bowl.
On low speed, add the flour mixture in 3 additions, alternating with the chocolate mixture, and beginning and ending with the flour, until just incorporated. Do not over mix. Scrape the bottom and sides of bowl with a rubber spatula to thoroughly combine.
Divide batter evenly among the cake pans, smoothing the tops. Bake until a skewer inserted in the center comes out clean, 15 to 18 minutes. Cool on a wire rack 15 minutes. Run a thin knife between the cake and the pan to loosen. Invert each cake onto a cooling rack and peel off the parchment, then re-invert onto a lightly greased rack. Cool completely.
To prepare the Cake:
Once the cakes have cooled completely, place them in the freezer for 10 minutes to chill slightly. Let the ice cream stand at room temperature until soft enough to spread, about 5 minutes.
Place 1 cake layer, top side up, on a serving plate. Working quickly, top with half of the ice cream, spreading it almost to the edges of the cake. Place another cake layer, top side up, onto the ice cream layer. Top with the remaining ice cream, spreading it almost to the edges. Set the remaining cake layer, bottom side up, on the second ice cream layer. Cover with plastic wrap and immediately place in the coldest part of the freezer for at least 2 hours or up to 2 days.
When ready to serve, prepare the whipped cream: In a small bowl, whisk together the powdered sugar and 1/2 cup of the heavy cream until thoroughly combined and no lumps remain. Gradually stream in the remaining cream until combined. Pour the cream mixture into the canister of a cream whipper. Follow the manufacturer's instructions to whip the cream.
Pipe the whipped cream over the top of the ice cream cake in a decorative fashion, such as curly-cues, rosettes or shell patterns. Sprinkle the remaining 1/4 cup crushed peppermint candies over the top of the cake.
To slice, dip a knife in hot water and wipe it dry before cutting each slice.
Serves 8 to 10.
Ice Cream Maker