Line two baking sheets with parchment paper and set aside.
Cream butter and sugar with an electric mixer until light and fluffy -- approximately 2 to 3 minutes.
Add egg, beating until combined.
Add egg yolk and vanilla then beat until combined. Set aside.
In a separate bowl, whisk together flour and salt.
Add flour mixture to wet ingredients then beat on low speed until just incorporated. Avoid overmixing.
Using a 1 Tablespoon-sized cookie scoop, form dough into evenly-sized balls.
Place cookie balls 1.5 inches apart on the parchment-lined baking sheets. Dip the bottom of a drinking glass into granulated sugar then flatten a cookie ball to 1/4 inch thickness. Continue dipping the glass bottom into sugar and repeat the flattening process for the remaining cookie balls.
Using the backside of a 1/4 teaspoon measuring spoon, make an indent in the top of each cookie.
Place cookie sheets in the refrigerator and chill dough for 1 hour.
Preheat oven to 325 degrees then bake cookies for 12 - 14 minutes or until bottoms are golden and edges are set.
Remove baking sheet from the oven then deepen the thumbprint indents with the backside of the 1/4 teaspoon.
Allow cookies to cool on the baking sheet for 5 minutes then transfer cookies to a wire rack to cool completely before adding icing.