Preheat the oven to 375°F and adjust the rack to the upper-middle position. Melt 9 tablespoons of butter in a 12-inch cast-iron skillet over medium heat. Stir constantly until butter turns golden brown and emits a nutty aroma (about 5 minutes). Pour into a large bowl and mix in the remaining 3 tablespoons of butter until melted.

Whisk in the brown sugar, granulated sugar, vanilla, and salt until smooth. Add the egg and yolk, whisking until incorporated (about 30 seconds). Let the mixture sit for 3 minutes, then whisk again for 30 seconds. Repeat this resting and whisking process two more times until the mixture becomes thick, smooth, and glossy.

In a separate bowl, mix the flour and baking soda, then stir into the wet mixture until just combined (about 1 minute). Fold in the chocolate chips, ensuring an even distribution.

Wipe the cast-iron skillet clean with a paper towel, then transfer the dough back into it. Press the dough into an even layer. Bake for 20 minutes, rotating the skillet halfway through, until the edges are golden brown.

Remove the skillet from the oven and place it on a wire rack. Let the cookie cool for 30 minutes before slicing into wedges. Serve warm, optionally with ice cream.