Heat oven to 350 degrees.
Line a large baking sheet with parchment paper.
In the bowl of an electric mixer fitted with the paddle attachment, beat together 1 1/2 sticks butter, 1/2 cup confectioners sugar, 1/4 teaspoon salt, and 1 teaspoon vanilla on medium-high speed until smooth, about 2 minutes.
Beat in 1 1/4 cups flour, beginning on low speed and increasing to medium high until dough comes together into a ball.
Roll dough by teaspoonfuls into balls, and place 1 inch apart on parchment paper. Refrigerate tray for about 30 minutes to prevent cookies from spreading too much.
Bake at 350 degrees for 10 minutes, remove from oven, and use the bottom of a wooden spoon or a 1/4 teaspoon measuring spoon into tops of cookies to make indentations.
Return to oven, and continue baking until light brown on the edges, about 7 to 9 minutes more. Remove to a wire rack to cool.
While the cookies cool, make the chocolate filling.