Add the butter to a stand mixer with a paddle attachment, or large mixing bowl with a handheld mixer. Beat it for 1 to 2 minutes until smooth. 1 cup unsalted butter
Add in the powdered sugar, one cup at a time, mixing on low speed until each cup is mixed in completely. Scrape the sides and bottom of the bowl as needed. 3.5 cups powdered sugar
Add in the heavy cream (starting with 3 TBS), and the vanilla extract, peppermint extract, and salt. Mix on medium speed until everything is well combined. 3-4 TBS heavy cream, 1/2 tsp vanilla extract, 1/4 tsp peppermint extract , 1 dash salt
Mix the frosting for another 2 to 3 minutes at medium speed until the frosting is smooth and fluffy. Add a little more powdered sugar, or heavy cream as needed to get the right consistency.
Spread the frosting onto the cooled cookies, and sprinkle the tops with additional candy cane pieces.