Preheat oven to 350 degree F. Beat butter in large mixing bowl with electric mixer on medium to high speed for 30 seconds. Add granulated sugar and salt; beat until combined. Beat in vanilla. Beat in as much of the flour as you can with the mixer. Stir in any remaining flour with a wooden spoon. Stir in pecans.
Shape slightly rounded teaspoons of dough into balls the size of large grapes. Place 1 inch apart on ungreased cookie sheet. Bake in preheated oven about 12 minutes or until bottoms just begin to brown. Transfer to wire rack; cool. Sprinkle with powdered sugar. Makes about 72 cookies.