Preheat the oven to 375 degrees F (190 degrees C). Grease cookie sheets.
In a medium bowl, stir together the flour, white sugar, brown sugar, and baking soda.
Add the egg, vanilla, and butter; mix until dough forms. Stir in the chopped walnuts.
Roll dough into 1 inch balls and place 2 inches apart onto the prepared cookie sheets.
Bake for 8 to 10 minutes in the preheated oven, until the edges are golden brown. Allow cookies to cool on baking sheet for 5 minutes before removing to a wire rack to cool completely.
