Brown the butter, then cool until solid but spreadable.

Beat the butter and sugar until fluffy, add vanilla if using, then mix in the flour and salt until a dough forms.

Shape the dough into two logs, wrap in plastic, and chill until firm.

Preheat oven to 325°F (165°C). Slice into ¼-inch rounds, place on parchment-lined sheets, and bake 14 to 16 minutes until edges are just golden.

Sprinkle warm cookies with flaky sea salt if desired, cool on the sheet for 5 minutes, then transfer to a rack to cool completely.