Melt the butter in a heavy-bottomed saucepan over medium heat. Swirl the pan occasionally and continue cooking until the butter turns dark brown and smells nutty. Remove it from the heat and let it cool until nearly solid.
Preheat the oven to 350°F (175°C). Grease the sides of a 9-inch (23cm) cake pan and line the bottom with parchment paper. A springform pan makes removing the cookie cake even easier.In a large bowl, combine the browned butter, granulated sugar, and brown sugar. Beat with a hand mixer on medium speed until creamy. Add the egg, egg yolk, and vanilla extract. Beat until the mixture is smooth and fluffy.
Add the flour, cornstarch, salt, and baking soda. Mix on low speed just until combined. Be careful not to overmix the dough.
Fold in 3/4 cup of the chocolate chips.
Spread half of the cookie dough evenly over the bottom of the prepared pan. Drizzle 1/4 cup of caramel sauce over the dough.
Add the remaining cookie dough in small dollops over the caramel layer. Gently spread it out to cover the caramel as much as possible.Scatter the remaining chocolate chips over the top. Drizzle with 2 tablespoons of caramel sauce. Bake for 25–27 minutes, or until the center is set and the edges are golden brown. Let the cookie cake cool completely in the pan before removing it.
Beat the softened butter in a medium bowl with a hand mixer or stand mixer until creamy.
Add the powdered sugar, whipping cream, vanilla, and salt. Start mixing on low speed, then gradually increase the speed. Beat for about 5 minutes, or until the frosting is light and fluffy.
Once the cookie cake has completely cooled, carefully remove it from the pan.
Pipe the frosting around the edge of the cookie cake and add any decorative pattern you like. Drizzle the remaining caramel sauce over the top and finish with sprinkles.Slice into wedges and serve.