Mix flour, powdered sugar, and saslt in processor. Add butter, pistachios, egg yolk, and vanilla.
Using on and off turns, mix until moist ball forms. Transfer dough to work surface. Divide dough in half. Form each half into 8 X 1 1/4" log (if dough is too soft, chill until firm, about 30 minutes). Wrap logs in plastic, refrigerate until firm, about 4 hours. Can be made 5 days ahead, keep chilled.
Preheat oven to 325 degrees.
Slice logs into 1/4" rounds, rolling log every few slices to retain round shape. Place rounds on ungreased baking sheets, spacing 1" apart.
Bake shortbreads until barely golden, about 18 minutes. Cool shortbread on baking sheets.