Lightly grease a 9 1/2-inch round fluted tart pan with a removable bottom.
In a small saucepan over medium heat, melt the butter and continue to cook until it caramelizes and emits a nutty, butterscotch aroma. You will notice brown flecks in the bottom of the pan as the butter deepens in color. This should take about 3-6 minutes. Be careful not to burn the butter. Remove from the heat.
In a medium bowl, combine the melted butter with the brown sugar, vanilla and salt. Add the flour and mix until incorporated. Spread the dough evenly and up the sides of the prepared pan. Let stand for at least 2 hours or preferably overnight. (Do not refrigerate)
Preheat oven to 300°F.
Bake the shortbread for 45 minutes. Remove the pan from the oven. Let cool.