Preheat your oven to 350°F (175°C). Lightly butter a 9-inch ceramic pie dish and set aside.
In a small bowl, whisk together the melted butter, brown sugar, cinnamon, and chopped nuts, if using. Stir until a thick paste forms. Set aside.
In a medium bowl, whisk together the melted butter, brown sugar, and granulated sugar until smooth. Add the egg, egg yolk, and vanilla and whisk until fully incorporated.
Fold the flour, baking soda, and salt into the wet ingredients using a rubber spatula. Mix just until a cohesive cookie dough forms. Fold in the white chocolate or cinnamon chips, if using.
Press about two-thirds of the cookie dough evenly into the bottom of the prepared pie dish. Spoon half of the cinnamon mixture over the dough. Scatter the remaining cookie dough over the top in small pieces, then dot with the remaining cinnamon mixture.Run a butter knife through the dough in a loose figure-eight pattern to create swirls, being careful not to overmix the layers. Bake for 28–32 minutes, or until the edges are deep golden brown and the center is set around the edges but still soft in the middle. Be careful not to overbake—the center will continue to set as it cools.
While the cookie bakes, add the granulated sugar to a small saucepan over medium heat. Let the sugar melt, stirring occasionally with a heatproof spatula, until it becomes a smooth amber-colored caramel.
Once the sugar has melted, carefully add the butter a few pieces at a time, whisking continuously. The mixture will bubble vigorously.
Slowly pour in the heavy cream while whisking. Continue cooking for about 1–2 minutes, until the caramel is smooth and slightly thickened. Remove from the heat and stir in the vanilla and sea salt.
Let the caramel cool for a few minutes so it thickens slightly. Drizzle the warm caramel sauce over the cookie just before serving. Slice into wedges and serve warm with vanilla bean ice cream and extra caramel sauce.
