Beat the butter with an electric mixer until pale and fluffy. Add the powdered sugar and beat for 2 minutes, until well combined. Add the vanilla and combine.
In a separate bowl, combine the flour and salt. Add the butter mixture and mix until the dough sticks together when pinched. Shape into a ball, wrap in plastic wrap and chill in the fridge until firm, about 1 hour.
Roll the dough out on a floured surface to 1/8 inch thick. Use a circular 2-inch round cookie cutter to cut out rounds, then use a circular 1-inch round cookie cutter (or a bottle cap!) to cut out the centers.
Place on a baking sheet lined with parchment paper. Bake at 325F for 13-15 minutes. Cool on the pan for 15 minutes, then transfer to a wire rack and cool completely.