Place butter in a microwave-safe bowl and cover with parchment paper. Place a microwave-safe plate over the parchment paper to hold it down. Microwave butter for 5 minutes. Check butter; continue to microwave in 30-second intervals if needed, until butter is golden brown and has released a nutty aroma, an additional 30 to 90 seconds.
Mix flour, baking soda, and salt in a large bowl and set aside.
Beat brown butter, brown sugar, and sugar together in another bowl until smooth and creamy. Add vanilla and egg and mix until well blended. Add flour mixture and chocolate chips; mix until incorporated. Place dough in a container and refrigerate for a minimum of 4 hours to 24 hours.
Preheat the oven to 350 degrees F (175 degrees C). Line a baking sheet with parchment paper.
Scoop cookies with a medium-sized cookie scoop and set on the prepared baking sheet, 2 inches apart.
Bake in the preheated oven until lightly browned on the edges, 13 to 15 minutes. Cool on the baking sheet, then transfer to a wire rack.