Preheat oven to 400°. Combine flour and sugar. Cut in butter until mixture resembles cornmeal. Sprinkle with vanilla and 3/4 cup chopped walnuts. Assemble mixture with hands and form into a ball. Divide dough into two pieces and form each into a roll about 1 1/2" in diameter. Brush with egg white and roll in remaining chopped nuts. Cover with plastic wrap and refrigerate until firm.
Carefully cut into slices 1/4" thick and place on ungreased baking sheet. Bake for 10 minutes (or until golden brown). Remove from baking sheets and cool. Makes 6 dozen cookies.