To make the frosting, melt the white chocolate using a double boiler or in the microwave in 20 to 25 second intervals, stirring between each. Let it cool at room temperature for about 10 minutes so it is no longer warm.
In a stand mixer, beat the softened butter on medium speed for about 3 minutes, until very light and creamy. Add the cooled white chocolate and beat until fully incorporated.
Add the powdered sugar, heavy cream, peppermint extract, vanilla, and salt. Beat on medium-high speed for 2 to 3 minutes until fluffy and whipped. Add more cream as needed to reach a soft, spreadable texture.
Fold in the crushed candy canes.
Spread the frosting generously over the cooled brownies and finish with extra crushed candy canes on top. Refrigerate for 20 to 30 minutes to help the frosting set slightly, then slice into clean squares.