Preheat oven to 300 degrees. Lightly grease two 9-inch round cake pans.
In a medium-sized bowl, cream together the butter, salt, sugar, and vanilla, then beat in the flour. The dough will be stiff. Divide the dough in half (each half will weigh about 12 ounces), and press it into the prepared pans, smoothing the surface with your fingers or the bottom of a measuring cup. Prick the dough with a fork in an attractive pattern.
Bake the shortbread for 35 to 40 minutes, until it's golden brown around the edges. Remove it from the oven and loosen the edges with a heat-resistant plastic knife or table knife. Wait 5 minutes, then carefully turn the shortbread out onto a clean work surface, all in one piece.
Using a pizza wheel, baker's bench knife, or sharp knife, cut each round into 12 wedges. (Do this while the shortbread is still warm; if you wait until it's cool, it won't cut easily.) Transfer the wedges to a rack to cool.