Beat butter and sugar until light and fluffy, about 2 minutes.

Mix in sourdough discard and vanilla, then add flour and salt, mixing just until dough forms.

Divide dough in half, roll into 1 1/2-inch logs, wrap in plastic, and chill 30 minutes.

Preheat oven to 350°F (175°C). Slice logs into 1/4-inch rounds, place on parchment-lined sheets, and bake 12 minutes until edges are golden.

Cool on sheets 5 minutes, transfer to a rack, and dust with powdered sugar if desired.