Preheat the oven to 350˚ degrees F.
In the bowl of an electric mixer fitted with a paddle attachment, mix together the butter and 1 cup of sugar until they are just combined. Add the vanilla. In a medium bowl, sift together the flour and salt, then add them to the butter-and-sugar mixture. Mix on low speed until the dough starts to come together. Dump onto a surface dusted with flour and shape into a flat disk. Wrap in plastic and chill for 30 minutes.
Roll the dough 1/2-inch thick and cut with your favorite cookie cutter. Place the cookies on an ungreased baking sheet (mine is a baking pan with snowflake molds on it) . Bake for 20 to 25 minutes, or until the edges begin to brown. Allow to cool to room temperature.