Brown butter in a small saucepan over medium heat (for a stages of browning butter see link here). Remove from heat and pour and scrape browned butter and bits into the bowl of an electric stand mixer and allow to rest 30 - 35 minutes until cool.
Preheat oven to 350 degrees. In a mixing bowl, whisk together flour, baking powder, baking soda and salt. Stir in oats, set mixture aside.
Add light brown sugar and granulated sugar to cooled butter in bowl of electric stand mixer, and using a paddle attachment, whip mixture on medium speed, 3 - 4 minutes, frequently scraping down sides of bowl.
Blend in peanut butter. Add in egg and vanilla and mix until blended. Slowly add in dry ingredients and mix just until combined. Stir in chocolate chips.
Scoop dough out and shape into balls, about 3 Tbsp each (I did each one 50g by weight). Transfer to parchment paper lined baking sheet, spacing them at least 2-inches apart. Bake in preheated oven 10 - 11 minutes (if you make smaller cookies, reduce baking time).
Sprinkle tops lightly with sea salt if desired. Allow to cool several minutes on baking sheet before transferring to a wire rack to cool. Store in an airtight container.