In a medium sized saucepan set over medium heat, brown the butter. Stir the butter occasionally until it turns amber in color and develops little brown specks. Once you see the specks, immediately remove the butter from the heat.
You should be left with ~125g of butter. If you have a tiny bit less that's okay. Transfer out 45g of brown butter into a small mixing bowl and place it in the fridge to chill for 5 minutes. Transfer the remaining brown butter into another small mixing bowl and set it aside at room temperature.
Brown Butter Cookie Cake
Preheat the oven to 350°F / 175°C. Coat a 6" cake pan with butter and sugar.
Remove the butter from the fridge. Add in the brown sugar, granulated sugar, egg yolk, milk, and vanilla and whisk until well combined.
Add in the flour, baking powder, baking soda, and salt. Use a rubber spatula to fold the cookie dough until a few flour streaks remain. Then, fold in your chocolate chips just until no flour streaks remain.
Pat the dough down into your prepared cake pan. Top with additional chocolate chips if desired. Bake the cake for 20-25 minutes, or until the edges are golden brown and the center only slightly moves when you tilt the pan.
Let the cookie cake rest in the pan for at least 20 minutes before transferring it out to fully cool.
Brown Butter Buttercream
Once the cake has fully cooled to room temperature, make the buttercream. If your butter hasn't solidified at this point, you can place it into the freezer, stirring often until it solidifies.
Using a hand mixer or stand mixer fitted with the paddle attachment, beat together the butter and salt for 1-2 minutes, until the mixture is light and fluffy.
Slowly beat in the powdered sugar until all of it has been used. Add in the heavy cream and vanilla, then beat for an additional 2 minutes.
Once the cake has cooled to room temperature, transfer the buttercream to a piping bag. Frost the cake, serve, and enjoy!