Toast the Pecans: In a skillet, melt 2 tablespoons of butter over medium heat. Add the pecans and toast for about 5 minutes, stirring occasionally until they are golden and aromatic. Spread the pecans on a baking sheet to cool, then coarsely chop them.
Prepare Brown Butter: In a saucepan, melt the remaining 1 cup of butter over medium heat, stirring constantly until the butter turns golden brown and develops a nutty aroma (about 5-7 minutes). Remove from heat and let cool to room temperature.
Mix the Dough: In a large bowl, combine the cooled brown butter, both sugars, the egg, the yolk, and vanilla extract. Mix well with a spatula or hand whisk until fully incorporated.
Add Dry Ingredients: In the same bowl, sift in the flour, baking soda, and salt. Mix until just combined; avoid over-mixing.
Fold in the Toasted Pecans: Add the cooled, chopped pecans to the dough, folding gently to distribute them evenly.
Refrigerate the Dough (Optional): You can refrigerate the dough for at least 30 minutes to enhance the flavor and firm up the dough, though baking immediately is also fine.
Shape the Cookies: Using a #20 cookie scoop (about 3 tablespoons of dough per cookie), form balls of dough. Place on a tray and freeze for 5-10 minutes to help them hold their shape while baking.
Bake: Preheat the oven to 350°F (175°C). Place the cookies on a parchment-lined baking sheet, leaving space between each one. Bake for 12-15 minutes, until the edges are golden.
Cool and Serve: Let the cookies cool on the baking sheet for a few minutes before transferring them to a wire rack to cool completely.