Brown 2 cups of butter total following the method in the article notes. Let both portions cool and fully re-solidify.
Preheat oven to 350 degrees F. Grease and flour three cake pans and line the bottoms with parchment rounds.
Whisk the cake flour, baking powder, and salt in a medium bowl. Set aside.
In a stand mixer, beat the re-solidified brown butter with both sugars on medium speed for 3 minutes until light and fluffy.
Add eggs one at a time, mixing until just combined after each.
Mix in the vanilla extract.
With the mixer on low, add the flour mixture in three additions, alternating with the buttermilk in two additions, starting and ending with flour. Add the extra milk if the batter looks stiff.
Divide the batter evenly among the three pans and bake for 24 to 28 minutes, until a toothpick comes out clean.
Cool in the pans for 10 minutes, then transfer to a wire rack to cool completely before frosting.
Beat the re-solidified brown butter alone for 2 minutes until pale and creamy. Add powdered sugar one cup at a time, alternating with the heavy cream. Add vanilla and salt, then beat on high for 2 minutes until fluffy.
Frost between each layer, apply a thin crumb coat, refrigerate for 20 minutes, then apply the final coat of frosting, slice, and serve.