Adjust oven rack to upper-middle position and preheat oven to 350°F.
Butter a 12-inch cast-iron skillet with 1 tablespoon of butter and also spray it with nonstick cooking spray. In a medium-sized bowl, whisk together the all-purpose flour and baking soda; set aside.
13 tablespoons unsalted butter, 1¾ cups all-purpose flour, ½ teaspoon baking soda
Melt 9 tablespoons of butter in a heavy-bottomed, medium-sized saucepan set over medium heat. Brown the butter, swirling continuously, until the butter has melted and is dark golden brown, about 3-5 minutes. 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 bowl and add the remaining 3 tablespoons butter; stir until melted.
Add the 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.
1 cup dark brown sugar, ¼ cup granulated sugar, 2 teaspoons pure vanilla extract, 1 large egg, 1 egg yolk
Add the flour mixture and stir just until no flour is visible.
Add chocolate chips and stir just until incorporated.
½ cup semisweet chocolate chips, ½ cup milk chocolate chips
Press the cookie dough evenly into the prepared skillet and transfer to the oven. Bake for 23-28 minutes (rotating after 13 minutes), or until middle has barely set.
Cool 5-10 minutes before slicing into wedges.