In a small saucepan set over medium heat, melt the butter. Stir the butter occasionally until it turns amber in color and develops little brown specks.
Pour the brown butter into a medium sized mixing bowl. You should be left with at least 80g of brown butter (but not more than 85g.) If you have a little less than 80g, you can add in a few grams of water until you reach 80g.
Mix the peanut butter into the hot brown butter until the mixture is smooth, then set the mixture in the freezer for 5 minutes.
Once the brown butter has chilled a little bit, whisk in both sugars, then whisk in the egg and vanilla.
Add in the flour, baking soda, baking powder and salt. Use a rubber spatula to fold the mixture together just until no flour streaks remain.
Cover the bowl tightly with a lid or with plastic wrap, and set the dough to chill in the fridge for at least 30 minutes, and up to 2 days.*
Preheat the oven to 350°F / 175°C. Line a cookie sheet with parchment paper or a silicone baking mat.
Scoop out heaping 3 tbsp. scoops of dough and evenly space them out on the cookie sheet to allow room for spreading.
Bake for 10-12 minutes, until the edges of the cookies start to turn golden brown.
Immediately after taking the cookies out of the oven, take a bowl that is slightly larger than the cookies and swirl it around over the cookies to make them a little more uniform and circular.