Melt the butter in a light-colored saucepan over medium heat. Cook, swirling often, until amber with brown bits and a nutty smell, about 3 to 5 minutes. Pour into a large bowl to cool for 10 minutes.

Whisk the cooled brown butter with both sugars. Add the egg and vanilla, then beat in the peanut butter until smooth.

Fold in the flour, baking soda, and salt until no dry streaks remain, being careful not to overmix.

Roll the dough into tablespoon-sized balls and place on parchment-lined sheets. Press with a fork for crosshatch marks if you like.

Bake at 350°F (175°C) for 10 to 12 minutes, until edges are set and centers are puffy. Sprinkle with flaky salt warm, cool on the sheet 5 minutes, then transfer to a rack.