Brown the butter in a light-colored saucepan over medium heat until golden and nutty, about 5 to 7 minutes. Pour into a bowl and cool to room temperature.

Cream the cooled brown butter with 1 cup granulated sugar and brown sugar until fluffy, then beat in the egg and vanilla.

Stir in the flour, baking soda, and salt just until no dry streaks remain.

Press the dough into a disc, wrap tightly, and chill at least 1 hour.

Preheat oven to 350°F (175°C). Scoop rounded tablespoons of dough, roll in the extra sugar, and bake on parchment-lined sheets for 10 to 12 minutes until edges are set. Cool on the sheet 5 minutes, then transfer to a rack.