Preheat oven to 410° degrees F. Line 2 baking sheets with parchment paper or silicone liners.
In the bowl of a stand mixer, cream together the cubed butter pieces and granulated sugar for 30 seconds on low speed. Increase speed to medium and beat for another 30 seconds, then increase speed to high and beat for about 30 seconds or until fluffy and light.
Add in the eggs, beating well after each addition, followed by the vanilla. Add in the cocoa powder, cornstarch, baking soda, salt, cake flour, and all-purpose flour and mix on low speed until just incorporated. Stir in the peanut butter chips until combined.
Refrigerate the dough for 15 minutes.
Using your hands, scoop out 6 ounces of cookie dough (use a digital scale for accuracy). Roughly form the dough into a ball shape - it should not be compacted and smooth like a normal cookie dough ball. Place 3" apart staggered on a baking sheet, only 4 cookie dough balls per baking sheet.
Bake one sheet at a time in the middle rack of the oven for 8-10 minutes, up to 12 minutes, checking after the 8 minute mark. Cookies should be dry and dull on top and slightly underdone in the center; they will continue to cook and set up outside of the oven. Once cookies are finished, let sit on the hot baking sheets for at least 30 minutes before moving, storing, or eating for the gooey, melty center. Cookies will be very fragile straight out of the oven.