Preheat the oven to 375 degrees F (190 degrees C). Grease cookie sheets.

Stir flour, white sugar, brown sugar, and baking soda together in a large bowl until well combined. Add butter, egg, and vanilla; mix until dough forms. Stir in walnuts.

Roll dough into 1-inch balls and place 2 inches apart on the prepared cookie sheets.

Bake in the preheated oven until the edges are golden brown, 8 to 10 minutes. Let cookies cool on the sheets for 5 minutes before removing to a wire rack to cool completely.