Preheat oven to 350 degrees and spray two 6-inch cake pans generously with butter or oil. Separate 1/2 cup white chocolate chips and dark chocolate chips into separate bowls. Microwave one at a time in 30 second increments, stirring after each, until melted. Allow to cool about 5 minutes on the counter while you prepare the body of the cake.
In the bowl of a stand mixer or with a hand mixer, beat oil, yogurt, and sugar until soft and fluffy. Add eggs one at a time and blend until smooth, then add vanilla and milk. In a separate bowl, sift flour, baking powder, and salt. In two batches, add dry ingredients to wet and beat until mostly combined. Pour in hot water and whip once more. Some streaks of flour are okay!
Remove half of the batter into a separate bowl. Scoop dark chocolate and cocoa powder into one batch and stir until thoroughly mixed. In the second bowl, mix in melted white chocolate.
Pour batters separately into the two prepared cake pans. Bake for 40-45 minutes or until centers are set and knife comes out mostly clean with just a couple of crumbs. Cool in pans for 5 minutes, then turn onto a wire rack to cool completely.
When the cakes are thoroughly cooled, make the buttercream. Melt the other 1/2 cup of white chocolate chips and allow to cool for 5 minutes. Beat powdered sugar, butter, peppermint extract, and white chocolate until soft and fluffy, about 3-5 minutes. If you want a slightly stronger peppermint flavor, feel free to add a few more drops, but BE CAREFUL! Peppermint extract is potent.
Very gently cut each cake layer in half so you have four slabs of cake. I turned the cake slowly as I cut, making more shallow indentations, rather than trying to slice all the way through to get the most level division as possible. This worked well for me.
Place one dark chocolate layer on top of a plate or cake stand. Frost generously until the top of the layer is thoroughly coated, then add a layer of white chocolate. Frost. Add second dark chocolate layer. Frost. Finish with the final white chocolate layer. Swipe a thin layer of buttercream atop and all the way around the body of the cake. Refrigerate for about 10 minutes. This is your crumb layer and will prevent the knife from accidentally slicing bits of the cake off as you continue to decorate.
Once the crumb layer is set, finish off frosting the cake with the peppermint buttercream. Break large slabs of peppermint bark candy and arrange on top, then sprinkle any variety of nonpareils or Christmasy jimmies to finish. When ready, gobble that cake up!