Preheat the oven to 350 degrees.
In a medium bowl, cream butter, shortening, and sugar with an electric hand mixer or stand mixer. Beat in eggs.
In a separate bowl whisk flour, cream of tartar, baking soda, and salt together.
Combine flour mixture with wet ingredients.
Scoop cookies, using a medium cookie scoop, and place them on baking sheets lined with parchment paper. Completely cover each cookie dough ball with M&Ms. No need to cover the bottom of each dough ball, just the top and sides. You can do this by hand or press the cookie dough ball into a bowl of M&M candies.
Bake for 14 minutes. Let cookies stand on baking sheet for 5 minutes before transferring to a cooling rack.
Allow cookies to cool completely. To make an ice cream sandwich add a scoop of vanilla ice cream onto one cookie and press a second cookie on top. Press.