Preheat oven to 350°. Line a 9x13-in. pan with parchment paper or coat with nonstick spray.
In a large bowl, cream the butter and sugar until light and fluffy, then add the peppermint and mix. Combine the flour and cornstarch and gradually add to the butter until well combined.
Press shortbread evenly in the pan, then pierce a few times with a fork. Place in the oven and bake until lightly browned, about 40-45 minutes. Cut while warm. Cool completely before icing.
In a bowl, beat the confectioners sugar, butter, milk, peppermint and red food coloring until smooth. Spread evenly over the shortbread.
Melt the white chocolate in the microwave for 1 ½ to 2 minutes. Mix every 30 seconds until melted and smooth. Pour over the buttercream and smooth out.
Sprinkle crushed candy cane over the white chocolate. Let the chocolate harden before serving.