Preheat oven to 350 degrees. In a mixing bowl whisk together flour, oats, baking powder, baking soda and salt for 20 seconds, set aside.
In the bowl of an electric stand mixer fitted with the paddle attachment, cream together butter, brown sugar and granulated sugar. Mix in egg and vanilla, then blend in peanut butter.
With mixer set on low speed slowly add in dry ingredients and mix just until combined. Stir in semi-sweet and milk chocolate chips.
Scoop dough out about 2 1/2 Tbsp at time (using a cookie scoop or two spoons, one to scoop and the other to drop the dough. Or you can let it rest at room temp for about 15 minutes and you should be able to roll them into balls for more uniform shapes), and drop onto Silpat or parchment paper lined baking sheets.
Flatten cookies slightly then bake in preheated oven 9 - 10 minutes (they'll look slightly under-baked). Cool on baking sheet 5 minutes then transfer to a wire rack to cool completely. Store in an airtight container at room temperature.