In a large bowl cream the butter and vegetable shortening. Add the peppermint extract, salt, milk, and powdered sugar. Mix until completely incorporated.
Add the food coloring and mix until completely mixed in. Make sure to scrape the bowl to ensure the color is completed incorporated.
Preheat the oven to 350 degrees. Scoop 1 tablespoon of the cookie dough into the palm of your hand and form it into a ball. Place it on a cookie sheet lined with parchment paper.
Bake the cookies for 8 minutes. Allow them to cool for 10 minutes while still on the cookie sheet then transfer to a wired rack to completely cool.
Add the frosting into a piping bag with a round tip. Take 1 cooled cookie and pipe the frosting all around the edges moving into the center of the cookie. Take another cookie and place it on top. Squeeze down slightly to ensure the cookie sticks together. Continue this step until all sandwiches are made. Enjoy!