Soften the butter in the microwave (or on the stove top) until nearly melted, about 1 minute. Let cool slightly. Cream the butter and sugars with an electric mixer until blended. Add the egg and the yolk, mixing well. Add the vanilla and mix well. In a separate small bowl, combine the flour, baking soda, and salt. Add the dry ingredients to the butter mixture and beat until no flour streaks remain. Stir in the chocolate chips and walnuts.
Cover the dough with saran wrap and place in the refrigerator for at least 1 hour.
Preheat the oven to 350 degrees F.
Drop tablespoonfuls of dough on an ungreased cookie sheet (about 2 inches apart) and place in the oven. Bake for 11 minutes for soft cookies or 13 for crisp ones. The cookies should be slightly brown and puffy. Let the cookies sit on the cookie sheet for 10 minutes to cool before removing to a cooling rack or a plate. Eat one (or two or three) warm with a glass of ice cold milk. Life doesn’t get much better than this!