While the cookies are baking, start the frosting. Beat 1 cup of softened butter in a stand mixer with a paddle attachment (or hand mixer) until light and fluffy, about a minute.
Add the powdered sugar, salt, peppermint extract, vanilla extract, and 4 T. heavy cream, and mix on low speed until incorporated. Increase the speed, and beat until creamy and fluffy, adding more heavy cream as needed to get the right consistency. If it's too thin, add a bit more powdered sugar.
Note - You can adjust the peppermint extract to your taste. Start with a little--you can always add more. You want to avoid the toothpaste taste!
Unwrap the candy canes or peppermint candies, place in a plastic quart bag, and use a rolling pin to crush into pieces. You can also use a food processor if you have one.
Add a handful of the peppermint bits into the frosting and mix with a wooden spoon. Amounts are to your preference! Make sure you have enough reserved to sprinkle on top of the finished cookies.