Melt butter in a small saucepan. Let butter cool and pour into a large mixing bowl.
Add sugars and beat for several minutes, until mixture is creamy. Add eggs and vanilla. Beat for one minute, or until eggs dissolve into butter mixture.
Combine flour, baking soda, and salt in a separate bowl. Add butter mixture, stirring well. Add chocolate chips and walnuts if desired. Refrigerate cookie dough for two hours or until firm.
Preheat oven to 325 degrees F (165 degrees C).
Drop by two tablespoonfuls of dough, spaced 2 inches apart on the baking sheet. Bake for 10-12 minutes, or until the cookies are lightly browned and puffed up. Cool on cookie sheet until they are firm enough to put on a wire rack.