Preheat oven to 350 degrees Fahrenheit. Line 2 large baking sheets with parchment paper and set aside.
In a standing mixer with a paddle attachment, cream the butter and both sugars for about 1 minute until fluffy. Beat in the egg, vanilla extract, baking soda, and salt, and mix to combine. Once combined, fold in peppermint candy.
Scoop dough into 1-inch balls. Place on prepared baking sheets, leaving 3 inches between each cookie.
Bake 9-10 minutes Cookies will look underdone but this is what you want! The peppermint candy will harden them up after cooling.
To make the frosting, combine cream cheese and butter for about 2 minutes in a standing mixer. Cream until light and fluffy. Add in powdered sugar and vanilla extract and whip on high for an additional 30 seconds.
Sandwich the cookies with frosting and enjoy!
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