Preheat oven to 475°F.
In a food processor combine powdered sugar, 1 1/3 cups pecans, and flour. Process until the nuts become fully incorporated and are almost flour-like in consistency. Scrape down sides if necessary.
Add egg white, apricot jam, and espresso shot into flour mixture. Blend until all liquids are thoroughly incorporated and mixture forms a loose, dough-like mass. Remove nut dough from processor bowl, coat with a dusting of confectioners' sugar, wrap tightly with plastic wrap, and refrigerate until thoroughly cooled, about 30 minutes.
Divide nut dough into four equal pieces. Form each dough segment into 5 to 6-inch cylinders and roll in remaining 1 1/3 cups chopped pecans to coat surface. Roll nut-coated cylinders into 8 to 10-inch long ropes (no smaller than 1 inch in diameter) and cut crosswise into four equal baton pieces about 2 to 2 1/2 inches long.
Place each baton on a parchment-lined or greased cookie sheet and refrigerate for 1 hour or until dry.
Bake the baton cookies for 10 minutes or just until they begin to brown lightly. If batons begin to crack on the surface remove all from oven and allow to cool completely before serving or storing. Store in an airtight container for up to 3 days.