Place the butter in a medium saucepan over medium heat. Melt the butter, then continue to cook while whisking constantly. The butter will foam up and then begin to turn a golden brown color with small dark specks at the bottom. Once it smells nutty and looks toasted, immediately remove it from the heat and pour it into a large heat-proof bowl to cool for 15 minutes.
While the butter cools, place the chopped pecans in a dry skillet over medium heat. Toast them for 3 to 5 minutes, stirring frequently, until they become fragrant. Remove them from the pan immediately so they do not burn. This step is essential for releasing the oils and enhancing the nutty flavor.
Add the brown sugar and granulated sugar to the cooled brown butter. Use a whisk or electric mixer to combine them until smooth. Add the egg and vanilla extract, beating again until the mixture is light and creamy. This should take about 2 minutes to ensure the sugars are well incorporated.
In a separate bowl, whisk together the flour, baking soda, and salt. Gradually add the dry ingredients to the wet mixture, stirring with a spatula until just combined. Be careful not to overmix, as this can make the cookies tough. Fold in the toasted pecans gently at the end.
Preheat your oven to 350°F (175°C). For the best texture, chill the dough in the refrigerator for at least 30 minutes. Scoop rounded tablespoons of dough onto a parchment-lined baking sheet, spacing them 2 inches apart. Bake for 10 to 12 minutes or until the edges are golden brown. Let them cool on the pan for 5 minutes before transferring to a wire rack.