Preheat oven to 325°F (165°C). Line the bottoms of three 6-inch cake pans (or two 8-inch cake pans) with parchment paper, and grease the sides. Set aside.
Sift flour, baking powder, baking soda, and salt together in a medium bowl. Set aside.
In a large bowl, beat butter, oil, and sugar together until they are combined, about 3 minutes.
Add egg whites, one at a time, until each is mixed in before adding the next. Scrape down the sides of the bowl as needed.
Add almond and vanilla extracts and mix to combine. Add half of the dry ingredients and mix until almost combined. Add the buttermilk and mix.
Add the remaining half of dry ingredients, and mix until just combined. We don't want to overmix the batter or it will be dense.
Distribute the cake evenly between the prepared cake pans. Bake for 35-38 minutes, or until a toothpick inserted comes out clean.
Let cakes cool in their cake pans for 10 minutes at room temperature. Run a knife around the cakes to loosen them from the pans, then turn out onto a wire rack to cool to room temperature.