Line baking sheet with parchment paper or a nonstick baking mat; set aside.

Place butter and sugar in the bowl of a food processor; process until creamy. Add egg yolks, salt, and vanilla; process for 30 seconds until combined. Add 1 cup flour and process until smooth. Add remaining cup of flour to processor; process to combine.

Turn dough out onto work surface and form into flat square; wrap with plastic wrap and transfer to refrigerator until chilled.

Place a nonstick baking mat on work surface. Halve the dough and place one half on baking mat; top with a sheet of plastic wrap. Roll out dough between baking mat and plastic wrap until it is ⅛-inch-thick. Transfer to fridge to chill until firm, about 30 minutes. Repeat process with remaining half of dough.

Using a 2-inch round fluted cookie cutter, cut out dough and transfer to prepared baking sheets. Transfer baking sheets to refrigerator until dough is chilled, about 20 minutes.

Roll out scraps of dough, and repeat. Repeat process with remaining half of dough. Transfer to refrigerator and chill until firm, about 30 minutes.

Preheat oven to 325°F. Mix together remaining egg yolk with 1 teaspoon water; brush tops of cookies with egg white mixture. Sprinkle with sanding sugar, if desired.

Transfer to oven. Bake until cookies are pale golden, 12 to 15 minutes. Transfer to a wire rack to cool.