Cube the butter and place in a light-colored skillet. (Light colored helps you determine when the butter begins browning.) Melt the butter over medium heat, stirring constantly.
Once melted, the butter will begin to foam. Keep stirring. After 5-8 minutes, the butter will begin browning; you’ll notice lightly browned specks begin to form at the bottom of the pan and you'll begin to smell a nutty aroma. Once browned, immediately remove from heat, pour into a heat-proof bowl, and set aside for 10 minutes to cool slightly.
While the brown butter cools, preheat the oven to 350°F (180°C) and grease an 8×8-inch baking pan or line with parchment paper. Set aside.
To the brown butter (which should still be liquified), add the sour cream, milk, egg whites, vanilla extract, and vanilla bean paste and whisk until well combined. Add the sugar and whisk again until combined.
Sift in the flour, baking soda, baking powder, and salt, and mix until just combined.
Pour the batter into the prepared pan, and smooth out evenly with a spatula. Bake the cake for 28-30 minutes or until a toothpick inserted comes out clean and relatively crumb-free. Remove cake from oven and let cool completely to room temperature before frosting.