For the cake: Position a rack in the center of the oven and preheat to 350 degrees F. Grease three 6-inch cake pans (or a 9-by-12-inch rectangular pan) with butter. Cut parchment circles to fit into the pans, and then butter the parchment. Whisk together the cake flour, all-purpose flour, baking powder and salt in a medium bowl and set aside.
In a stand mixer fitted with the paddle, cream together 1 3/4 cups of the sugar and the stick of butter on medium speed until light and fluffy, about 5 minutes. Turn down the speed to medium low and slowly drizzle the oil into the creamed butter. Add the yolks one at a time, then the whole eggs, mixing after each. Add the vanilla and blend. Alternate adding the flour mixture and buttermilk into the batter, starting and ending with the flour.
In a separate, very clean bowl, whisk the 2 egg whites until foamy, then slowly sprinkle the remaining 1/4 cup sugar over them as you whip the whites to medium peak. Fold the whites into the cake batter.
Divide the batter evenly among the pans. Bake until a tester comes out clean, 30 to 40 minutes. Allow to cool completely before removing from the pans.
For the brown butter icing: In a saucepan, melt the butter. Cover the pan and allow to cook over medium-low heat until the butter starts to turn golden brown, about 5 minutes. Strain the browned butter through a fine-mesh sieve. Allow the butter to cool to room temperature, about 1 hour (this can be done the day before and left out at room temperature).
In a stand mixer fitted with the paddle, cream the brown butter with the powdered sugar. Add the vanilla and salt. If the icing is too thin, add more powdered sugar. If it is too thick, add a tablespoon of heavy cream or milk.
Using a sharp, serrated knife, cut off the top crust of the cake layers. Slice each layer into 2 even layers. Cover the first layer with a 1/8-inch-deep layer of icing. Cover with blueberries and raspberries and then cover with another 1/8-inch-deep layer of icing. Repeat this with another layer of cake, icing, berries and icing, then top with a third layer of cake. Repeat these steps to assemble the second cake.
Ice the outsides of the cakes and then decorate the tops of the cakes with berries. Serve immediately, or refrigerate for up to 24 hours. (Allow the cakes to come to room temperature before serving.)