Sift the all-purpose flour, cake flour, baking powder, baking soda, sugar, and salt into a medium bowl.  Using your fingers or a pastry cutter, work the cold butter into the flour until the pieces of butter are pea-sized.  Add the buttermilk and, with your hands or a rubber spatula, stir until the mixture is just beginning to come together to form a shaggy dough. Be careful not to over mix.

Sprinkle a generous amount of all-purpose flour on a work surface and turn the dough out of the bowl onto the floured surface. Sprinkle the top of the dough with a bit more flour. Use your hands to gently press the dough to a rough rectangle about 1/2-inch thick. Fold one-third of the dough up and over onto itself, then repeat with the remaining third of the rectangle (like you’re folding a letter in thirds). If the dough is too loose to hold together as you fold it, use a bench scraper to help lift and support it while folding. Pat the dough again to a ½-inch thick rectangle and repeat the folding process. Using a floured rolling pin, gently roll the dough to a final rectangle shape about ¾-inch thick. Using a large chef’s knife dipped in flour, trim the edges to even them up, then cut the dough into either 6 or 8 equal portions (cleaning and flouring the knife between each cut). Be sure to cut straight down when cutting the dough - a twisting motion will prevent the dough from rising. Arrange the biscuits with at least an inch between them on the prepared baking sheet, and brush with the melted butter. Transfer the biscuits to the refrigerator to rest for at least 15 minutes while you preheat the oven to 450 degrees F. 

Place the biscuits in the center of the oven and cook until they are golden brown and flaky, 14 to 16 minutes. 

Serve hot with butter or honey butter.