Melt butter in a saucepan over medium heat, stirring occasionally, until golden with brown specks and nutty aroma, about 8-10 minutes.
Pour browned butter into a bowl and cool to just above room temperature (about 93°F/34°C), roughly 30 minutes.
Beat cooled brown butter with both sugars until smooth and slightly fluffy, 1-2 minutes.
Add egg and vanilla, beat until creamy and well combined.
Whisk flour, baking soda, and salt in a separate bowl, then stir into butter mixture until just combined.
Fold in chopped pecans until evenly distributed; dough will be stiff.
Shape dough into a 9-10 inch log on plastic wrap, wrap tightly, and refrigerate for 1 hour until firm.
Preheat oven to 350°F (175°C) and line baking sheets with parchment.
Slice chilled log into ½-inch rounds and place 2 inches apart on prepared sheets.
Bake 10-12 minutes until edges are golden but centers still look slightly underbaked.
Cool cookies completely on baking sheets on a wire rack.
Melt white chocolate in 20-second microwave bursts, then drizzle or dip tops of cooled cookies.
Let chocolate set at room temperature or in the refrigerator for 15-20 minutes before serving.