Make the cookies: Heat oven to 350 degrees. In the bowl of an electric mixer fitted with the paddle attachment, beat together 1 stick butter, sugar, salt, and vanilla on medium-high speed until smooth, about 2 minutes. Beat in flour, beginning on low speed and increasing to medium high.
Roll dough by teaspoonfuls into balls, and place 1 inch apart on an ungreased baking sheet. Bake for 10 minutes, remove from oven, and press thumb or other rounded tool (they will be hot) into tops of cookies to make rather large indentations. Return to oven, and bake until light brown on the edges, 5 to 7 minutes more. Remove to a wire rack to cool.
Make the filling: Combine chocolate and butter in a small heat-proof bowl. Microwave for 30 second intervals until completely melted and smooth, making sure to stir thoroughly in between (mine only took 30 seconds).
Fill the warm or cool thumbprints with the chocolate mixture.