Cream butter and brown sugar in the bowl of a stand mixer fitted with the paddle attachment. (You can also use a hand mixer.) Mix in egg and vanilla until smooth. Add pudding mix, baking soda, and salt. Mix until combined. Mix in flour, then stir in sprinkles and M&Ms.
Scoop 2 tablespoon balls of cookie dough onto a cookie sheet covered with parchment or a silpat baking mat. There’s no need to space them out, you’re going to chill them. Cover and chill for at least 30 minutes.
Preheat oven to 350°F. Line cookie sheets with parchment paper or silpat baking mats. Place chilled cookie dough balls 2” apart on cookie sheet. Bake 9-11 minutes or until the edges just start to turn brown. Cool 5 minutes on cookie sheet before transferring to a rack to cool completely. Store in an airtight container for up to 4 days or freeze for up to one month.