Preheat the oven to 325 degrees F, mist two 9-inch cake pans with non-stick spray, and line them with circles cut from parchment paper.
Place the cake flour, all-purpose flour, sugar, baking powder, baking soda, and salt in the bowl of an electric mixer, and stir on low speed to combine.
Add the butter, mixing on medium-low speed until the mixture resembles damp sand (about 1 minute).
Add the eggs, one at a time, scraping the bottom and sides of the bowl to be sure all of the ingredients are fully incorporated.
Add the yogurt, milk, and vanilla.
Mix on medium speed for 2 to 3 minutes, to aerate the batter and build the cake's structure.
Divide the batter equally between the prepared pans, and bake for 22 to 28 minutes, or until a toothpick inserted in the thickest part of the cake comes out clean or with a few moist crumbs.
Cool completely before filling and glazing.