Preheat oven to 350°F. Spread pecans out into a single layer on a sheet pan. Bake for 10-15 minutes until fragrant.
Set pecans aside to cool. If using pecan halves, give half of them a rough chop so they incorporate into the dough better.
Increase oven temperature to 375°F. Line a sheet pan with parchment paper.
In a small saucepan set over medium heat, melt the butter. Continue to cook the butter until it turns golden brown, stirring frequently, about 3 minutes. Remove from heat and let cool for a few minutes.
Pour browned butter into a large bowl and add the sugar, brown sugar, and vanilla. Stir with a wooden spoon until combined. Add the egg and egg yolk, stirring for 30 seconds after each. Let batter rest for about 5 minutes, then stir vigorously for about 30 second. Batter should lighten in texture and color.
In a separate bowl, combine the flour, baking soda, and salt. Add to the cookie batter all at once and stir until combined. Add pecans when a few streaks of dry ingredients remain and mix until incorporated.
Drop dough by rounded tablespoon onto the prepared sheet pan. Bake until cookies turn golden around the edges but centers aren't completely set, about 9-11 minutes. (If baking larger cookies, bake for 11-13 minutes.) Let cool 5 minutes on the pan, then transfer to a wire rack to cool completely.