Preheat the oven to 350 degrees F. Spray a 10-inch cast-iron skillet with nonstick cooking spray.
Prepare the cake mix according to the package's egg white method. Add the peppermint extract and mix to combine. Pour the cake batter into the prepared skillet. Bake until the cake is set and a toothpick inserted comes out clean, 35 to 45 minutes. Let the cake cool until the top of the cake is cool to the touch.
Pulse the candy canes in a food processor until you get pea-sized candy, 5 to 6 times. Add 1/2 cup of the crushed candy canes to the Frosting and fold to combine. Frost the cake with the Peppermint Frosting, then sprinkle the remaining crushed candy over the entire cake. Serve slices with scoops of peppermint ice cream.
Buttercream Peppermint Frosting:
In the bowl of a stand mixer fitted with a paddle attachment (or using a hand mixer), beat the confectioners' sugar together with the butter until well combined. Add the heavy cream and continue to beat until extremely light and fluffy. Beat in the peppermint extract until combined.