Preheat the oven to 350 degrees F.
Grease two 9-inch round cake pans, then use the extra flour to dust the greased pans.
In a bowl, whisk the remaining 3 1/4 cups cake flour, the baking powder, and the salt together.
In a second bowl, add 1/2 cup of the butter, the oil, and the granulated sugar together and use an electric hand mixture on medium speed to beat until it is smooth and combined, about 2-3 minutes.
Add the eggs, 1 at a time, to the butter mixture and beat well, scraping down the sides of the bowl after each addition.
Add 1 tablespoon of the vanilla extract to the butter mixture and beat to combine.
Alternate adding the flour mixture and the buttermilk to the butter mixture, beginning and ending with the flour mixture. Beat until the flour mixture is fully mixed in. Do not overmix.
Evenly divide the batter between the prepared cake pans and smooth the tops.
Bake the cakes until they are golden-brown on top and a toothpick inserted into the centers come out with a few moist crumbs, about 30-35 minutes.
Transfer the pans from the oven and let the cakes cool for 5 minutes.
Turn the cakes onto the rack and let them cool completely.
In the bowl of an electric stand mixer, add the remaining butter and the confectioners' sugar and mix on low speed; then increase to medium speed until combined.
Add the heavy cream and the remaining vanilla extract to the sugar mixture and mix on low speed until it is combined, about 1-2 minutes.
Turn the mixer off and add the cocoa powder to the frosting mixture; mix on low speed until it is combined.
Increase the speed to medium-high and mix the frosting until it is light and fluffy, about 5-7 minutes.
Place 1 of the cooled cakes on a cake stand.
Use an offset spatula to spread a layer of the frosting over the top of the cake.
Place the remaining cooled cake on top of the frosting layer.
Spread the frosting over the top and the sides of the cakes until they are covered and smooth.
Slice and serve.