Preheat your oven to 410 degrees F.
Using your stand mixer fitted with the paddle attachment, cream together cold cubed butter and both sugars for about 3 to 4 minutes, or until creamy. (after the butter is more incorporated after 2 minutes or so, I raised the stand mixer speed a little).
Add in your eggs, one at a time, mixing well after each addition. Add in the vanilla extract and mix again.
Using a separate mixing bowl, whisk together the dry ingredients: both flours, cornstarch, baking soda and salt. With the stand mixer on low, slowly add in all of the dry ingredients into the wet. Mix until just combined. Add in the chocolate chips and walnuts and mix one more time.
Scoop out dough into large balls and place on a baking sheet. To get the Levain Bakery style cookies, you want to make the cookie dough balls pretty large. Test a few to see the size you prefer. Bake right away, OR make all of your cookie dough balls and refrigerate for one hour. Test both ways to see which cookie you prefer. They both taste wonderful. The refrigerated ones have a slightly deeper flavor.
Place desired number of cookies on baking sheet. (I wouldn't bake more than 4 at a time.) Bake for about 11 to 12 minutes or until golden brown on the bottom edges and browned slightly on the top. Remove from the oven and let cool for a few minutes. Enjoy!