Preheat oven to 350 degrees F. Grease four 8-inch round cake pans and line the bottom with parchment paper; set aside. In a large mixing bowl beat butter and sugar for about 2 minutes.
In a medium bowl sift together the flour, baking powder, and salt; set aside.
In another medium bowl combine egg whites, milk, and vanilla.
Add 1/3 cup of the flour mixture to the butter mixture, beat to combine. Add half of the milk mixture to the butter mixture. Continue to alternate adding flour and milk mixtures to the butter mixture.
Pour batter evenly into three of the prepared pans. Bake 25 to 30 minutes or until a toothpick inserted near center comes out clean. Cool in pan on a wire rack for 5 minutes. Turn out cake onto rack, remove parchment paper, and let cool completely. Cakes may need to be leveled to remove darker edges or doming.
Melt creme de menthe baking chips in a microwave-safe bowl on HIGH for 30-second intervals until melted (about 1 minute total time). Stir until smooth.
Beat butter and sugar at medium speed in a stand mixer, beating about 2 minutes or until well blended. Add eggs, 1 at a time, beating just until blended after each addition. Add cooled melted chocolate, beating just until blended.
Sift together flour, baking soda, and salt. Gradually add to chocolate mixture alternately with sour cream, beginning and ending with flour mixture. Beat at low speed just until blended after each addition. Gradually add hot water in a slow, steady stream, beating at low speed just until blended. Stir in peppermint. Spread batter into remaining pan.
Bake at 350 degrees 35 to 40 minutes, or until center is firm. Cool on a wire rack. Cake can be leveled so it's the same height as the white cake layers. Place cake layers in the freezer for at least 30 minutes and up to 2 hours before frosting
When ready to assemble, prepare Peppermint Buttercream Frosting. Remove cakes from freezer. Place chocolate cake on cake stand and cover top with 1/3 cup of buttercream. Place white cake on top and cover in 1/3 cup of frosting. Repeat with remaining layers. (If cakes start to appear crumbly or too soft, place back in the freezer for 30 minutes.)
Cover outside of cake in buttercream. place cake on rotating cake stand. Using a knife or large offset spatula, press knife gently into the base of the cake and start turning the cake stand. Do not move the offset spatula, only turn the cake. Go around and around, slowly moving offset spatula up and eventually over the top of the cake. Continue swirling pattern around top of cake. Top the edges of cake with additional creme de menthe baking chips.
Beat butter in a stand mixer with the paddle attachment on medium speed for 1 minute. Add 3 cups of powdered sugar and turn your mixer on the lowest speed (so the sugar doesn't blow everywhere) until the sugar has been incorporated with the butter. Increase mixer speed to medium and add peppermint extract, salt, and 5 to 6 tablespoons of milk. Beat for 3 minutes. Slowly add remaining sugar. If your frosting needs to be thinned out, add remaining milk 1 tablespoons at a time.