Melt the butter in a light-colored saucepan over medium heat. Continue cooking, swirling occasionally, until the butter turns a rich golden brown and smells nutty. This should take about 5-8 minutes. Remove from heat immediately and pour into a heatproof bowl to stop the cooking. Stir in the chopped pecans and set aside to cool slightly.
In a large bowl, whisk together the brown sugar and granulated sugar. Add the eggs one at a time, whisking well after each addition. Stir in the vanilla extract and the cooled brown butter pecan mixture.
In a separate medium bowl, whisk together the flour, baking soda, and salt. Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Do not overmix.
Cover the dough and refrigerate for at least 30 minutes, or until firm enough to scoop.
Preheat your oven to 350°F (175°C). Line baking sheets with parchment paper.
Scoop rounded tablespoons of dough onto the prepared baking sheets, leaving about 2 inches between cookies.
Bake for 10-12 minutes, or until the edges are golden brown and the centers are still slightly soft. They will continue to set as they cool.
Let the cookies cool on the baking sheets for a few minutes before transferring them to a wire rack to cool completely.
Notes
For an extra layer of flavor, consider adding a sprinkle of flaky sea salt on top of the cookies right after they come out of the oven.