Preheat the oven to 300 degrees F (150 degrees C). Line 2 baking sheets with parchment paper.

Combine cold butter, white sugar, lemon zest, salt, and vanilla in a bowl; slowly stir together until butter has softened somewhat and ingredients come together.

Add flour to the butter mixture. Flour your hands and, using fingertips, rub mixture together until buttery crumbs form, 1 to 2 minutes. Press mixture into a ball.

Transfer dough to lightly floured work surface and press into a 1/2-inch- thick disk. Cover with a piece of plastic wrap. Roll dough to 1/4-inch thickness.

Cut dough with a 2-inch round cutter, rerolling scraps if needed. Arrange cutouts 1 inch apart on the prepared cookie sheets. Use a sharp knife to score 1/4- to 1/2-inch decorative lines around the outer edges of each cookie, cutting toward the center. Use a toothpick or skewer to poke small holes at the end of every other line near the center.

Bake in the preheated oven until light brown around the edges, 22 to 25 minutes. Transfer to a wire rack to cool completely.