Pulse the toasted pecans in a mini food processor until they resemble a coarse meal (do not over process).
In the bowl of an electric mixer fitted with the paddle attachment, cream together the butter, brown sugar, and granulated sugar until smooth. Add the vanilla and salt and mix on low just until combined. Mix in the chopped nuts and the flour, just until the flour is completely incorporated.
Divide the dough into two equal pieces and set one half aside. Roll dough between two pieces of plastic wrap until about 1/4-inch thick. Repeat with the remaining dough. Place on a baking sheet and refrigerate for at least 20 minutes, or up to two days. (The dough can also be frozen at this point until ready to bake; thaw in the refrigerator before using for at least 2 hours).
Cut out cookies using a 2-inch cookie cutter and place on a cookie sheet lined with parchment paper (or lightly greased baking sheets), rerolling any scraps. Refrigerate until the cookies are firm, about 10 minutes.
Preheat the oven to 300° and position the racks in the middle of the oven. Brush each cookie very sparingly with egg white. Press one pecan half into each cookie and sprinkle lightly with sugar.
Bake just until the cookies are just beginning to brown, 20 to 25 minutes, rotating between upper and lower racks after 10 minutes. Carefully transfer cookies to a cooling rack and cool completely. They will keep in an airtight container for several days.