In small saucepan, combine the powdered sugar, ¼ cup butter, and 3 tablespoons corn syrup; bring to a boil. Remove from heat. Stir in the pecans. Refrigerate at least 10 minutes. (This can be done while the cookie dough chills.)
MAKE THE COOKIES:
In large bowl, use an electric mixer to beat 1 cup butter and sugar until light and fluffy. Add ½ cup corn syrup and egg yolks; blend well. Stir in the flour; mix well. Cover with plastic wrap- refrigerate 2 hours for easier handling.
Preheat the oven to 375°F. Line baking sheets with parchment paper or silpat mats.
Shape the dough into 1½-inch balls. Place them 2-inches apart on the prepared baking sheets. Bake for 5 minutes. Remove from the oven. Brush the tops of the partially baked cookies lightly with egg white. With a small spoon, carefully scoop about ¾ teaspoon of the filling into the center of each cookie, pressing down slightly.
Return to the oven and bake an additional 6 to 9 minutes, or until light golden brown.
Cool for 2 minutes and then remove the cookies from the baking sheets to fully cool.
Notes
To toast pecans, preheat oven to 400 degrees F. and place the pecans on a small baking sheet. 3 to 5 minutes in the oven should be enough to get them a little bit toasted. Much longer than that and they'll start to burn!