In a large mixing bowl, cream butter, brown sugar, and granulated sugar until light and fluffy, about 3 minutes. Add peanut butter and beat until fluffy again, about 2 minutes, scraping down the bowl as needed. Beat in egg and vanilla until combined.
Sprinkle the baking soda and salt over the top. With the mixer on low, gradually add the flour, mixing just until the flour disappears. Slowly beat in the oats. By hand, stir in the M&Ms, chocolate chips, and peanut butter chips. If time allows, chill dough for 30 minutes or overnight.
When ready to bake, place racks in the upper and lower thirds of the oven and preheat the oven to 350 degrees F. Line two cookie sheets with parchment paper or a silicone baking mat. Drop cookies onto the sheets by rounded tablespoons. Bake for 10 minutes in the upper and lower thirds of the oven, until lightly golden brown (cookies may appear underdone). Let rest on cookie sheet for 5 minutes, then transfer to a wire rack to cool. Serve with a big glass of cold milk.
Store leftover Monster Cookies in an airtight container for up to 1 week or freeze for up to 3 months. Let thaw overnight in the refrigerator, then transfer to the counter.
To freeze unbaked cookies: portion the dough into balls and arrange on a parchment-lined baking sheet. Place the sheet in the freezer until the dough is firm, then transfer to an airtight container or freezer-grade ziptop bag and freeze for up to 3 months. Bake directly from frozen, adding to the baking time as needed.