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Preheat the oven to 375 F.
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Place 9 tablespoons of butter in a small saucepan and melt it over medium heat. Cook the butter, stirring constantly, until it’s dark golden brown and the bubbling has subsided, about 4-5 minutes.
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Pour the brown butter in a medium heat-safe bowl. Add the remaining 3 tablespoons of butter and stir until incorporated.
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Add the brown sugar, granulated sugar, vanilla, and salt and whisk until smooth. Whisk in the egg and egg yolk.
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Let the butter mixture rest for 3 minutes, then whisk for 30 seconds. Repeat this two additional times. The resulting dough should be thick, smooth, and shiny.
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Add the flour and baking soda and stir until combined. Stir in the chocolate chips.
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Place the Nutella in a microwave safe bowl and heat briefly in the microwave so that it is slightly melted and easy to spread.
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Press half of the cookie dough into a 10 or 12 inch cast iron skillet.
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Spread the Nutella onto the cookie dough. Spread the remaining cookie dough on top of the Nutella layer.
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Bake cookie for 20-25 minutes, until the cookie is golden brown and the edges are set.
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Let the cookie cool in the pan for 30 minutes before serving.