Sift together flour and salt. Set aside. Toast pecans in a small pan over medium heat until fragrant and lightly browned. Set aside to cool completely then chop into small pieces.
In the bowl of an electric mixer fitted with the paddle attachment, beat butter and sugar until smooth. Scrape down the sides then mix in the egg yolks one at a time on a slow speed, until each is just combined. Stir in vanilla extract.
Scoop the dough out shape it into one-inch balls. Place them on a flat surface and press your thumb or use bottom of a one-teaspoon measuring spoon to make an indentation.
Making sure the thumbprint cookies don’t touch, transfer to a tray and refrigerate for 90 minutes.
Preheat oven to 350 degrees and transfer cookies to a baking sheet lined with parchment paper or a silicone baking mat. Leave two inches between the cookies, fitting twelve on a pan at a time. (Keep any remaining cookies chilled until ready to bake.) Bake for 14-15 minutes, or until the bottoms are golden brown.
Optional: Once the cookies are out of the oven, use a measuring spoon to deepen the indentation.
Let the cookies cool for 5 minutes in the pan before transferring to a wire rack to cool.
Brown Butter Icing
In a small bowl, mix together melt the brown butter. Stir in vanilla extract and salt. Sift the powdered sugar and beat vigorously with a fork until the icing is smooth.
Use a spoon to drizzle about one teaspoon of icing into the indentation of each thumbprint cookie. Sprinkle with toasted pecans and serve.
If you make the indentation while the cookies are on the tray, you make it more likely that the pressure will cause the cookies to stick to the it.
When the cookies are done baking, the indentation in the middle may seem a little shiny and undercooked. It will finish cooking during the cooling process.
Once the icing is done being mixed, work quickly to fill each of the thumbprint cookies. The icing will harden as the brown butter cools.
Store the iced thumbprint cookies in an airtight container for up to a week. Keep them in a single layer to preserve the brown butter icing.