Preheat oven to 350 Farenheit.
Prepare a 9"x9" cake pan by lining with parchment paper and a bit of oil spray.
Start by melting your butter in a saucepan over medium heat, giving it a stir as it melts. Once the butter is completely melted, it will start to bubble and sizzle while the water evaporates. Once the water has mostly evaporated, the milk solids toast and turn brown. It is helpful to whisk often at this stage, because the toasty bits can burn without movement. Keep cooking the butter until it has reached a nice golden brown with toasted brown bits. Pour butter into a large, heat proof bowl.
Add granulated sugar and salt to the brown butter, whisking well.
Add eggs, whole milk, vanilla, and milk to the butter mixture and whisk until fully combined.
Gently stir in the baking soda and flour with a spatula.
Pour batter into prepared pan and bake for 28-35 minutes or until a toothpick comes out clean. Let cool completely.