Preheat the oven to 350℉. Line two baking sheets with parchment paper or silpat mats.
In a large bowl, use an electric mixer to combine the butter, powdered sugar, salt and vanilla. Beat until smooth- about 2 minutes. Beat in the flour until combined.
Form balls using about 2 teaspoons of dough for each. Place balls 1 inch apart on the prepared baking sheets. You should be able to fit 15 cookies on each sheet. Bake 10 minutes, then remove from the oven and use your thumb to press a deep indentation into the middle of each cookie. Return to the oven and bake 7 to 9 minutes more, until light brown on the edges. When the cookies come out of the oven, check the middles. Use your thumb to press down a little bit in the middle of each cookie once again, being careful not to press too hard. Cool completely.
In a medium glass bowl, combine the chocolate, butter and corn syrup. Set the bowl over a pan of simmering water and use a whisk to combine the chocolate mixture until melted and smooth. Set the bowl aside for about 10 minutes or so to cool (and let it thicken a little). Fill the middles of the thumbprint cookies with small spoonfuls of the chocolate mixture. Let them sit on a rack to firm up. Then store them in a covered container for up to 3 days (or freeze them). That's it!