Brush two cookie sheets lightly with oil; dust lightly with flour, and set aside.
Beat egg whites (at room temperature) and salt in a large mixing bowl until soft peaks form. Gradually add 1/3 cup sugar, 1 tablespoon at a time, beating until stiff peaks form. Set aside.
Combine egg yolks, 1/8 cup sugar, and vanilla in a small mixing bowl; beat at medium speed of an electric mixer 5 minutes or until thick and lemon colored. Gently fold yolk mixture into egg whites just until blended. Gradually fold in cake flour, 1/8 cup at a time, sifting flour over surface of egg mixture before folding.
Gently spoon mixture into a pastry bag fitted with a 1/2-inch round tip. Pipe mixture onto prepared cookie sheets in three 1/2-inch-long fingers, 2 inches apart. Sprinkle remaining 1/3 cup sugar lightly over ladyfingers, using a fine-meshed sieve. Bake at 350° for 8 minutes or until lightly browned.