Preheat the oven to 350 F. In a medium bowl, whisk together flour, oats, salt, and baking soda. In a large bowl, mix together butter, peanut butter, granulated sugar and brown sugar until combined. Add in the egg and vanilla extract until smooth.
Add in the flour mixture, mix until it is fully combined. Fold in the chocolate chips and M&Ms. Use a cookie scoop to place the cookies onto a large ungreased baking sheet at least 2-inches apart. Bake for 10-12 minutes. Let cool completely.
In a large bowl, mix together butter, peanut butter, powdered sugar, and vanilla extract until smooth. Spread frosting onto half of the cookies, and sandwich with the remaining half.
