Preheat the oven to 350°F (175°C).
In a bowl of an electric mixer cream the butter well. Add the 1/2 cup of sugar gradually and cream the mixture until light and fluffy. Combine the flour and the cornstarch. Add the dry mixture gradually to the creamed mixture, mixing well after each addition. Turn the dough out onto a lightly floured surface and knead until smooth.
Divide the dough in half. Roll out each half into an 8-inch round between 1/4-inch and 1/3-inch in thickness. With a large spatula put the rounds on an ungreased baking sheet and with a ruler or a sharp spatula mark each round into 8 wedges. Press the tines of a fork decoratively around the edges, sprinkle each round with 1 tablespoon of the remaining sugar, and bake in the lower third of the oven for 35 minutes, or until lightly golden and cooked through. While the shortbread is still warm cut it into wedges with a pastry wheel or large sharp knife. Transfer to racks to cool.
Coconut Shortbread: Reduce the 1/2 cup of sugar to 1/4 cup and add 1/2 cup sweetened flake coconut to the dough. Knead the coconut into the dough and bake as directed above.