Preheat oven to 375°F. Sift together flour, baking soda, and salt. Set aside.
In a medium bowl, combine the butter, peanut butter, ½ cup white sugar, and brown sugar and mix until smooth. Beat in egg, then stir in vanilla. Gradually blend in the sifted ingredients.
Shape dough into 1-inch balls. Roll in remaining sugar. Place 2 inches apart on ungreased cookie sheets.
Bake in preheated oven for 10 to 12 minutes. Cool slightly, and remove from pan.