Add butter to a medium sauce pan and brown. Once browned let cool.
In a medium bowl, whisk the sugars with the cooled butter. Then add the egg, egg yolk, and vanilla. Whisk until sugar is dissolved and the mixture is smooth.
Using a spatula, fold the dry ingredients into the wet ingredients, until all the flour has been incorporated. Then fold in the chocolate and walnuts (if using).
Scoop and/or roll into 1-1.5oz balls (for regular) or 6oz balls for Levain copy cat. Place on a sheet tray and chill in the fridge for 2-24 hours.
Preheat oven to 350 for regular and bake on middle or lower third rack. 1oz cookies will take about 11 minutes, 1.5oz cookies will take about 16 minutes. About half way through pull the cookies out and bang the baking sheet on the stove top a few times.
Remove from the oven and let the cookies cool for 3-5 minutes.
Preheat oven to 425 for Levain and bake for 12-18 minutes