Preheat an oven to 350°F . Place a rack in the middle of the oven. Grease a 10" cast iron skillet (or square baking dish) with butter, and set aside.
Warm butter in a small saucepan over medium heat. The butter will melt, then foam and bubble, then start to brown. You will notice little brown bits forming at the bottom of the pan. Stir frequently during this process and keep your attention on the pot. The butter can go from wonderfully brown to burnt in just a few moments. You will know the butter is done when it gives off a nutty aroma and turns a golden brown.
Add brown butter to a large bowl with brown sugar, whisking to combine. Whisk in milk, sour cream, eggs, and vanilla until smooth. Add flour, sea salt, and baking powder to the bowl, stirring just until combined. Do not overmix here.
Pour the cake batter in the prepared cast iron skillet, and cook until the top turns light golden brown, and a toothpick inserted into the center of the cake comes out clean (25 - 30 minutes).