In the bowl of a stand mixer, cream together the butter, brown sugar and white sugar with the paddle attachment on medium speed for about 2 minutes, until fluffy. Beat in the egg and vanilla. Lastly, beat in the cornstarch, baking soda, salt and flour until a soft dough has formed. Stir in the chocolate chips and walnut pieces gently.
Refrigerate the cookie dough for AT LEAST one hour, up to overnight. This is MANDATORY and chilling the dough is necessary to produce thick, fat, chewy cookies that do not spread. Don't skimp on this step!
Preheat oven to 350 degrees F. Line 2 cookie sheets with silicone liners or mist the sheets lightly with cooking spray. Portion 1/4 cup sized balls of dough (or two Tablespoon-sized cookie dough scoop's worth of dough) into huge balls of dough and place them onto the baking sheet. Bake for approx. 12 minutes, rotating pans halfway through baking time. Cookies will be lightly golden on top and may appear slightly undone; this is normal as they will continue to set up as they cool. Do NOT over-bake them! Cool the cookies on the baking sheet for about 5-10 minutes before carefully transferring to a wire rack to cool completely.
Serve cookies warm or room temperature, if desired. For the full Levain Bakery effect, serve them warm!
**TO TOAST WALNUTS: preheat oven to 350 degrees F. Line a baking sheet with foil and place the walnut pieces on the baking sheet in an even layer. Toast for about 5-10 minutes, turning walnuts every so often, until fragrant and nutty. Keep a close eye on them as they can burn quickly. Allow to cool before stirring into dough.**