1. Adjust oven rack to upper-middle position and heat oven to 375 degrees. Melt 9 tablespoons butter in 12-inch cast-iron skillet over medium heat. Continue to cook, stirring constantly, until butter is dark golden brown, has nutty aroma, and bubbling subsides, about 5 minutes; transfer to large bowl. Stir remaining 3 tablespoons butter into hot butter until completely melted.
2. Whisk brown sugar, granulated sugar, vanilla, and salt into melted butter until smooth. Whisk in egg and yolk until smooth, about 30 seconds. Let mixture sit for 3 minutes, then whisk for 30 seconds. Repeat process of resting and whisking 2 more times until mixture is thick, smooth, and shiny.
3. Whisk flour and baking soda together in separate bowl, then stir flour mixture into butter mixture until just combined, about 1 minute. Stir in chocolate chips, making sure no flour pockets remain.
4. Wipe skillet clean with paper towels. Transfer dough to now-empty skillet and press into even layer with spatula. Transfer skillet to oven and bake until cookie is golden brown and edges are set, about 20 minutes, rotating skillet halfway through baking. Using potholders, transfer skillet to wire rack and let cookie cool for 30 minutes. Slice cookie into wedges and serve.