Preheat oven to 350F. Line 3, 8-inch cake pans with parchment paper and spray sides with non-stick baking spray. Can also use 2, 9-inch pans.
In a large bowl, cream together the butter and sugar until light and fluffy. Add the egg whites and peppermint oil. Whisk the mixture until egg whites are fluffy. Then add the remaining liquid ingredients: coffee and milk. Make sure coffee is cooled to room temperature.
In a separate bowl, mix together the dry ingredients: flour, cocoa powder, baking powder and salt. Using a fine mesh strainer/sifter, sift the flour into the cake batter in small batches, mixing after each addition just until combined.
Divide the cake batter evenly between the prepared pans. I recommend using a kitchen scale for accuracy. Bake in preheated oven for approximately 40 minutes or until a toothpick inserted into the center comes out clean. Remove cakes onto cooling racks to cool completely. Once cooled, use a long, serrated knife to level off the tops of cake layers if needed.
For Peppermint Buttercream:
Prepare the buttercream: place softened butter into mixer bowl and whisk on high speed for 5 minutes, until very light and fluffy. Add the peppermint oil and gradually add the confectioner's sugar, mixing well after each addition. Once all sugar is added, continue whisking the buttercream for another 3 to 4 minutes, until very fluffy. Reserve 2 cups of white buttercream in a separate bowl, then add red food coloring to the remaining buttercream.
Assembling the Cake:
To assemble the cake: place the first cake round onto a cake platter and soak the layer with Kahlua coffee liquor. Then add a generous amount of red frosting, spreading it to the sides with spatula. Repeat with second cake layer but use white frosting for the middle layer. Add third layer and frost top and sides of cake. Use a flat spatula to even off the tops and sides.
Use a spoon and your fingers to press crushed candy canes into the sides of the cake, if desired. Transfer the remaining white frosting into a pastry bag tipped with a star tip and create rosettes on top of the cake. Top with mini candy canes. Keep cake refrigerated; remove 30 minutes prior to serving.