Adjust oven rack to upper-middle position and heat oven to 350°F.
Butter a 12-inch cast-iron skillet with 1-tablespoon butter and also spray it with nonstick cooking spray. In a medium-size mixing bowl, whisk together all-purpose flour and baking soda; set aside.
Melt 9 tablespoons butter in a heavy-bottomed medium-size saucepan over medium heat. Brown the butter, swirling it continuously (3-5 minutes) until butter has melted, and is dark golden brown. Most of the bubbling will have subsided and the butter will have a nutty aroma. Watch very carefully the last minute – it can burn quickly. Transfer the melted butter to a large, heat-proof mixing bowl and add the remaining 3 tablespoons butter; stir until melted.
Add sugars, salts and vanilla; whisk until incorporated. Add egg and egg yolk and whisk until smooth, about 30 seconds. Let mixture stand 3 minutes. Repeat the process of whisking (30 seconds) and resting (3 minutes) for 3 times more times (a total of 4 times). Mixture will be thicker, smooth and shiny.
Add the flour mixture and stir just until no flour is visible.
Add chocolate chips and stir just until incorporated.
Press the cookie dough evenly into the prepared skillet and transfer to the oven. Bake at 350°F for 23-28 minutes (rotating after 13 minutes) or until middle has barely set.
Cool 5-10 minutes before slicing into wedges.
Serve & enjoy!
Makes 12 inch skillet cookie, 8-16 slices.
Take care not to burn the butter when you brown it or your cookie will taste bitter.
Don't over mix the dough or the cookie won't be nice and light. The flour should be just incorporated.
Bake the cookie in a fully pre-heated oven, and rotate it half way through cooking so that it cooks evenly.
Let cool for 5 minutes before slicing and serve with scoops of ice cream.