In a large bowl, sift together the cake flour, baking powder, and salt, then set aside.
Using a handheld or stand mixer with a paddle attachment, beat the butter on high speed until it becomes smooth and creamy, about 1 minute. Add both granulated and brown sugars, then continue mixing on high speed for about 3 to 4 minutes until the mixture looks fluffy and well combined. Remember to scrape down the sides and bottom of the bowl to make sure everything is mixed evenly.
Reduce the speed to low and add the eggs, one at a time, followed by the egg yolks. Then mix in the vanilla extract. Increase the speed to medium-high and beat until everything is fully incorporated, scraping the bowl again as needed.
With the mixer on low speed, add the dry ingredients in three parts, alternating with the buttermilk. Mix only until each addition is just combined. Avoid overmixing. If the batter has any large lumps, gently whisk them out. The texture should be slightly thick.
Scoop out 1 cup of the yellow batter and place it in a medium bowl. Melt 4 ounces of chocolate in the microwave, pausing every 20 seconds to stir until smooth. Add the melted chocolate to the reserved batter and stir until blended.
Pour an even layer of the plain yellow batter into each prepared cake pan. Spoon the chocolate batter on top, then finish with the remaining yellow batter. Use a knife to gently swirl the two batters together for a marbled effect. It doesn’t need to be perfect—the swirls will create a beautiful pattern as it bakes.
Bake the cakes for 25 minutes, or until a toothpick inserted into the center comes out clean. Mine are usually done around the 24-minute mark. If the tops start browning too quickly, loosely cover them with aluminum foil.
Once baked, remove the cakes from the oven and let them cool completely in their pans on a wire rack.