Preheat oven to 350 degrees. Line a baking sheet with silicone mats or parchment paper.
Cut cold butter into tablespoons. Add to stand mixer and mix on low until butter has somewhat softened.
Add sugars to mixer and mix on low until sugars and butter are well combined.
Add egg and vanilla to mixer, mix on low until there are no longer any lumps.
Sift together the dry ingredients. Add to mixer, mix on low. Scrape sides of bowl until well combined.
Measure dough mounds into 3 oz (small cookies) or 5 oz (jumbo cookies) and roll into round balls. (If cookie dough is too soft, let dough refrigerate for 30-60 minutes. Warm dough will make cookies fall flat) Place cookie dough balls 2-3 inches apart from one another on a lined baking sheet.
Bake for 10-12 minutes. Cookies will start to crack on top and look less wet when they're all done.
Let cookies cool on cookie sheets for at least 15 minutes before removing. Cookies must be completely cool before frosting!
For the Buttercream:
In a microwave safe bowl, melt the white chocolate chips and heavy cream for 30 seconds, stir until there are no remaining lumps.
Place butter in a stand mixer and beat on high until light and fluffy.
Add the white chocolate/cream mixture, salt, vanilla, peppermint extract, and heavy cream and mix until well combined.
 Slowly add the sifted powdered sugar while mixing on low until all sugar has been added.
Turn mixer to high and mix for 1-2 minutes until the buttercream is nice and fluffy.
Assembly
Frost cooled cookies with a generous amount of buttercream. Top with crushed candy cane or sprinkles.
Notes
*If buttercream has a yellow tint, it's most likely because of the vanilla extract you used. If this bothers you, you can use a little bit of purple (and just a TINY bit) food coloring gel to correct the color to be more white!