Preheat oven to 350 degrees F. Line 22-24 paper liners among 2 muffin cups. Set aside.
In a large bowl, combine the cake mix, egg whites, water, oil, and dry pudding powder and beat with a handheld electric mixer until moistened and combined, about 2 minutes. Divide the batter evenly among the muffin cups, filling about 3/4 full.
Bake for approximately 15-18 minutes or until a toothpick inserted near the center of the cupcakes comes out clean or with moist, not wet, crumbs. Cool the cupcakes completely.
For the peppermint mousse: in a small bowl, combine the peppermint crunch baking chips and the 1/4 cup heavy whipping cream. Microwave for 30 seconds on HIGH power, stirring until smooth. Set aside to cool off briefly. Meanwhile, in the bowl of a stand mixer, beat the remaining 3/4 cup heavy whipping cream on HIGH power for about 5-7 minutes or until soft peaks form. Pour in the melted peppermint chip mixture and continue beating on HIGH power until stiff peaks form.
Using a paring knife or cupcake corer, core out the centers of the cupcake, careful not to pierce through the bottom. Discard the cores. Fill a piping bag with the peppermint mousse and pipe the mousse into each muffin cavity.
For the frosting: In a small bowl, combine the white chocolate chips and the heavy whipping cream and microwave on HIGH power for 30 seconds, stirring until melted and smooth. In the bowl of a stand mixer, beat the softened butter until creamy and fluffy, about 1 minute. Add in the melted white chocolate and the vanilla and mix well. Lastly, add in the milk and confectioners' sugar and mix on low speed until fluffy and combined.
Pipe or spread the white chocolate frosting evenly over the cupcakes, taking care to cover the peppermint mousse filling completely with icing. Garnish with crushed candy canes and peppermint bark squares, if desired. Store any leftover cupcakes in the fridge, covered completely, for about 4-5 days or at room temperature for 2-3 days, airtight.