Preheat the oven to 350°F. Lightly coat three 6 inch cake pans in baking spray and line the bottoms with parchment rounds. Set aside.
In a large bowl, sift together the flours, pudding mix (if using), baking powder, baking soda, and salt. Whisk to combine.
In a separate bowl, combine sugars, milk, water, oil, 2 ounces of melted brown butter, vanilla, and eggs. Use an immersion blender or a whisk to combine the wet ingredients.
Pour the wet ingredients into the dry ingredients. Mix gently until just combined. A few small lumps are okay.
Use an electric scale to divide the batter between the three pans. You should end up with roughly 15- 16 ounces of batter in each pan. Bake 35- 40 minutes. The cakes should spring back when poked, a skewer should come out clean, and they should be starting to pull away from the edge of the pan.
Cool cake layers in the pans for about 10 minutes. Invert onto a cooling rack and cool completely. Wrap each layer in cling wrap and place in the fridge for at least one hour, or even overnight.