In a large mixing bowl, cream the softened butter and the 1/2 cup granulated sugar together until light and fluffy, 2 to 3 minutes. Beat in the vanilla if using.

Add the flour and salt and mix on low speed just until a soft dough forms and no dry flour remains. Stop as soon as it comes together — overmixing makes shortbread tough instead of delicate.

Gather the dough into a flat disk, wrap it well, and refrigerate for 30 to 60 minutes, until firm but still pliable.

Preheat the oven to 350°F (175°C). Line a baking sheet with parchment paper.

On a lightly floured surface, roll the chilled dough to 1/4-inch (6 mm) thickness, turning it as you go so it rolls evenly.

Cut into round scalloped cookies (about 2½ inches across) and place them on the prepared baking sheet, a little apart. Prick the tops with a fork for the classic shortbread look.

Roll the edges of each cookie lightly in the coating sugar (or sprinkle coarse sugar over the tops instead, as the original notes suggest).

Bake for 12 to 15 minutes, until the edges are just turning lightly golden and the centers stay pale. Do not let them brown deeply — pale is what keeps them tender.

Cool on the baking sheet for 5 minutes, then transfer to a wire rack and cool completely before serving or storing.