1. Preheat oven to 350 degrees F. Line a baking sheet with parchment paper or a silicone baking mat and set aside. If you are not using parchment or a silicone baking mat, you do not need to grease the baking sheet.
2. In the bowl of an electric mixer fitted with a paddle attachment, beat together 1 stick of the butter, sugar, salt, and vanilla on medium-high speed until smooth, about 2 minutes. Beat in the flour, beginning on low speed and increasing to medium-high.
3. Using a small cookie scoop or heaping teaspoons, roll the dough into balls, and place 1-inch apart on a baking sheet. Bake for 10 minutes, remove from oven, and make an indentation into the middle of each cookie. Return to oven, and bake until light brown on the edges, 7 to 9 minutes more. Remove to a wire rack to cool.
4. Combine chocolate, 2 tablespoons butter, and corn syrup in a small heat-proof bowl. Set over a pot of simmering water; stir occasionally until melted and smooth. (Alternately, you can microwave the mixture on 50% power in 30-second increments, stirring after each, until completely melted and smooth.) Allow to cool slightly. When cookies are cool, fill the thumbprints with the chocolate mixture.