Preheat the oven to 350°F. Prepare three 6-inch or two 8-inch cake pans by spraying the sides with a cooking spray (Baker's Joy is my favorite) and fitting a parchment paper circle to the bottom of the pan.
Sift the cake flour and then measure by spooning and leveling it in your measuring cup. Add the cake flour, baking powder, baking soda, and salt into a bowl and whisk to combine. Set aside.
In the bowl of your stand mixer (a handheld mixer works fine too!), beat the butter on high for two minutes until it's light and creamy. Add in the sugar and continue to mix on medium-high for another two minutes. Turn the mixer to low and add the eggs and egg whites, one at a time, mixing until just combined and scraping down the bowl and paddle as needed. Add the vanilla and sour cream, turn the mixer to high, and beat for one full minute. It will look curdled at this point, but don’t worry – it will become smooth cake batter at the end.
Add in the dry ingredients all at once and turn the mixer to low speed. Mix until just combined, then slowly pour in the milk. Continue to mix on low speed until just combined, about 30 seconds. Scrape down the sides and bottom of the bowl and give it a few stirs with a whisk to make sure there are no lumps. The batter will be slightly thick.
Pour the batter evenly into the prepared cake pans. Dollop the blackberry purèe by the spoonful on top of each cake and use a butter knife to swirl it into the batter, then bake for 30-35 minutes. They're done when they spring back to the touch and a toothpick inserted into the middle comes out clean. Let the cakes cool in the pan for five minutes before turning them out onto a wire rack for an additional few hours of cooling. Make sure they're entirely room temperature before applying any frosting.