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 fluffy.
Add in the egg and vanilla extract and mix until smooth. Add in the flour mixture, mix until it is fully combined. Fold in the M&Ms and chocolate chips. 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.
Place ice cream onto half of the cookies and sandwich with the remaining cookies. Store in the freezer until serving.