Preheat oven to 325°.
Lightly spoon flour into dry measuring cups; level with a knife. Combine flour, sugar, baking soda, and salt in a large bowl. Combine peanut butter, vanilla, eggs, and egg whites in a medium bowl, stirring well with a whisk; add to flour mixture, stirring just until blended.
Turn dough out onto a lightly floured surface; shape dough into a 10-inch-long roll. Place roll on a baking sheet coated with cooking spray; flatten to 1-inch thickness. Bake at 325° for 35 minutes. Remove roll from baking sheet; cool 10 minutes on a wire rack. Reduce oven temperature to 300°.
Cut roll diagonally into 18 (1/2-inch) slices. Place slices, cut sides down, on baking sheet. Bake at 300° for 20 minutes. Turn cookies over; bake an additional 20 minutes (cookies will be slightly soft in center but will harden as they cool). Remove from baking sheet; cool completely on a wire rack.