In a medium skillet over medium heat, melt butter, swirling pan occasionally until butter is a deep golden brown, about 3-5 minutes of cooking. Remove butter from heat, pour in a large bowl, and allow to cool 15 minutes.
Mix both sugars, peanut butter, egg, and vanilla into cooled brown butter until smooth. In a separate bowl, whisk flour, baking soda, and salt.
Fold flour mixture into butter until dough is just combined. Wrap dough in plastic wrap and refrigerate 2 hours or up to 6 days.
When ready to bake, preheat oven to 350F and line 2 large sheet pans with parchment paper. Roll dough into 1-inch balls, roll in additional sugar, and place 1 inch apart on prepared sheet pans. Flatten cookies slightly with the palm of your hand or a fork.
Bake cookies at 350F 10-12 minutes until edges are golden brown. Allow cookies to cool 10 minutes on sheet pan then transfer to a cooling rack to cool completely. Enjoy!