Line three 8-inch round cake pans with parchment circles and lightly grease the sides. Preheat the oven to 345°F (175°C).
Melt the white chocolate (double boiler or microwave at 50% power in 30-second bursts) and set aside to cool until just slightly warm to the touch; it should be fluid but not hot.
Roughly chop the peppermint bark and crush the candy canes for finishing; set both aside.
Whisk together the flour, baking powder, baking soda, salt, and sugar in a large bowl. In a separate pitcher, whisk the eggs, egg yolk, vanilla, sour cream, and oil until smooth.
Bloom the cocoa: whisk the cocoa powder into the very hot coffee until glossy and lump-free. Let stand 2 minutes to cool slightly.
Make the batter: add about half of the egg mixture to the dry ingredients and whisk to moisten. Pour in the cocoa-coffee mixture, whisking to combine, then add the remaining egg mixture and whisk just until you have a smooth, pourable batter. Do not overmix.
Divide batter evenly among the pans (about 1 3/4 cups per pan or weigh for accuracy). Bake 23–27 minutes, rotating pans once at 18 minutes, until a toothpick comes out with a few moist crumbs.
Cool cakes in pans 15 minutes, then turn out to racks and cool completely. For easiest decorating, chill layers in the freezer for 15–20 minutes before frosting.
While cakes bake/cool, make the buttercream: wipe a metal mixer bowl and whisk with vinegar or lemon juice to remove any grease. Add egg whites and sugar; set the bowl over a pot of barely simmering water and whisk constantly until the mixture reaches 160°F/71°C and feels smooth when rubbed between fingers (no sugar grains).
Transfer the bowl to a stand mixer and whip on high until the meringue is glossy, holds stiff peaks, and the bowl is cool to the touch, 8–10 minutes. Switch to the paddle, then add the butter a few pieces at a time on medium speed until thick and silky. If it looks soupy, chill the bowl for 5–10 minutes and continue mixing.
Beat in vanilla, peppermint extract, and a pinch of salt. Pour in the cooled melted white chocolate and mix just until fully incorporated and smooth, 1–2 minutes.
Level any domed cake tops. Set the first layer on a board with a dab of buttercream underneath. Spread about 3/4 cup frosting over the layer and sprinkle with 1/3–1/2 cup chopped peppermint bark. Top with the second layer and repeat. Place the final layer on top, cut-side down for the flattest finish.
Apply a thin crumb coat to seal in crumbs and chill the cake for 20 minutes. Add a final, thicker coat of buttercream and smooth the sides and top. Press crushed candy canes around the bottom edge for a festive border.
For a candy-cane swirl, streak the inside of a piping bag with a line of red gel, fill with remaining buttercream, and pipe rosettes or swirls on top. Finish with Andes Peppermint Bars tucked into the rosettes.