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Preheat the oven to 350°F with the rack in the center position.
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Place the walnuts in a small skillet and toast them over medium heat, stirring frequently to prevent burning, about 4 minutes or until the nuts smell fragrant. Transfer the walnuts to a small plate to cool while assembling the other ingredients.
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Working by hand in a large bowl or in the bowl of a stand mixer fitted with the flat beater, combine the sugars and melted butter. Once combined, add the egg, egg yolk, and vanilla and mix to combine. Scrape the sides of the bowl.
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Add the flour, salt, and baking powder to the bowl and stir until almost combined. A few streaks of flour are OK. Scrape the sides of the bowl.
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Add the chocolate chips and nuts to the bowl and stir until no flour streaks remain.
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Transfer the dough to a well-seasoned 10" cast iron skillet and spread into an even layer.
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Bake the skillet cookie for 25 to 30 minutes, or until it is evenly golden brown across the top and pulls away from the sides of the skillet slightly. A digital thermometer inserted in the center should read 195°F to 200°F when done.
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Remove the skillet cookie from the oven and cool on a wire rack for 10 minutes before serving directly from the skillet. (For a sliceable cookie, cool for 25 to 30 minutes in the skillet. Slice using a dull knife, like a nylon spreader or butter knife, to prevent damaging the skillet.)
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Storage information: Store any leftover skillet cookie in an airtight container at room temperature for up to 3 days. Reheat individual portions in the microwave, if desired.