Step 1Heat the oven to 350 degrees.
Step 2Place the butter in a small saucepan over low heat. Tilt the pan so the butter melts evenly. Pour the melted butter into an 8-inch square pan or 8-inch pie plate so the bottom is completely covered.
Step 3Place the eggs in a mixing bowl and, using a fork, beat them until they are all one color and look well blended.
Step 4Add the sugar, flour, baking soda, salt and vanilla, and, using a large spoon, beat the cookie batter until completely blended and smooth. Stir in the walnuts.
Step 5Pour the batter into the buttered plate or pan and bake until lightly browned, the dough feels firm to the touch and a toothpick inserted in the middle comes out clean, not damp and sticky, 15 to 20 minutes. Because ovens can vary in temperature, check the dough at 15 minutes.
Step 6Remove from the oven and let cool in the pan about 10 minutes. Remove from the pan to a wire rack to cool completely. Cut into wedges or squares. These freeze well if wrapped in two or three layers of plastic wrap.