Preheat oven to 350°F (177°C). Line two large baking sheets with parchment paper or silicone baking mats and set aside.
In a large mixing bowl, whisk together the flour, old-fashioned rolled oats, baking soda, and salt. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, or in a large mixing bowl using a handheld mixer, cream together the butter, brown sugar, and granulated sugar for about 1-2 minutes or until well combined. Mix in the peanut butter, egg, and vanilla extract until fully combined, making sure to stop and scrape down the sides of the bowl as needed.
Slowly mix in the dry ingredients until just combined. Add the M&M's and chocolate chips and gently fold them into the cookie dough.
Using a 1.5 tablespoon cookie scoop, scoop balls of cookie dough onto the prepared baking sheets, making sure to leave a little room between each one.
Bake in separate batches at 350°F (177°C) for 10-12 minutes or until the tops of the cookies are set. Remove from the oven and allow to cool on the baking sheet for 5-10 minutes, then transfer to a wire rack to cool completely.