Preheat oven to 350F. Line two baking sheets with parchment paper. In a bowl of an electric mixer fitted with the paddle attachment, beat together 1 cup (2 sticks) butter, confectioners sugar, salt, and vanilla on medium speed until smooth, about 2 minutes. Beat in flour, beginning on low speed and increase to medium until combined.
Form balls using 2 teaspoons of dough for each; place balls 1 inch apart on prepared baking sheets. Bake 10 minutes, remove from oven, press thumb into cookies to make deep,wide indentations. Rotate pan, and return to oven; bake until light and brown on edges, 7 to 9 minutes more. Transfer to a wire rack to cool completely.
Combine chocolate, remaining 6 tbsp. butter, and the corn syrup in a small heatproof bowl set over a pot of simmering water; stir occasionally until melted and smooth. Allow to cool a bit until slightly thickenerd. Fill thumbprints with the chocolate mixture, and set aside to firm up. Cookies can be stored in single layers in airtight containers at room temperature up to 3 days.