Preheat oven to 415°F. Line baking sheets with parchment paper.
In the bowl of a stand mixer with a paddle attachment, add the cold butter, brown sugar, and granulated sugar. Cream together until light and fluffy, 3-5 minutes. (You can also use a handheld mixer.)
Reduce mixing speed and add the eggs, one at a time, mixing well after each egg has been added. Scrape down the sides of the bowl as needed.
With the mixer on low, add the cocoa, cake flour, all-purpose flour, cornstarch, baking soda, and salt. Mix until the ingredients are incorporated.
Fold in the peanut butter chips and chocolate chips.
Using a ¼ cup (or 2 scoops of dough with a 2 tablespoon scoop), add the dough out onto a lined cookie sheet, about 2 inches apart. (I usually put 6 cookie dough balls onto each cookie sheet.) Use your hands to shape and slightly flatten each ball of dough. (If you used 2, 2 tablespoon scoops, combine the scoops of dough into one large cookie.)
Bake for 8 minutes. Let cookies rest for about 10 minutes before transferring to a wire rack.