Preheat the oven to 350 degrees F.

In a large bowl, using a hand-held mixer or stand mixer with a paddle attachment, cream together the butter and both sugars on medium speed, until light and fluffy. Mix in eggs, one at a time, mixing well after each addition and scraping the sides of the bowl as needed. Add vanilla extract and mix well.

In a separate bowl, stir together the flour, pudding mix, baking soda and salt. Slowly add the flour mixture, a little at a time, to the butter mixture, mixing just until combined. Carefully, fold in 1 cup mini M&M's.

Drop dough, by rounded tablespoonfuls, 2 inches apart onto an ungreased baking sheet. Press 4-5 of the remaining M&M's on top of each cookie. Bake 8 to 10 minutes, or until lightly browned and set in the middle. Let the cookies cool on the baking sheet for 10 minutes before moving to a cooling rack to cool completely.

Once cookies have cooled completely, remove ice cream from freezer and let sit at room temperature for 1-2 minutes to soften slightly. Working quickly, use an ice cream scoop to scoop some ice cream onto the flat side of a cookie. Top ice cream with a second cookie, making sure the flat side of the cookie is towards the ice cream. Press down gently to allow the ice cream to spread slightly. Immediately roll the edges of the ice cream sandwich in additional M&M's candies or sprinkles, if desired. Wrap ice cream sandwiches individually in plastic wrap and freeze until the ice cream is very firm, at least 4 hours.