In a large bowl, sift together the flour, cake flour, baking powder, baking soda and salt. Whisk together. Set aside.
In a separate large bowl or the bowl of a stand mixer, beat together the cubed butter, brown sugar and granulated sugar until combined and no lumps of butter remain. If you only have a hand mixer, this will take some time.
Mix the eggs and vanilla into the butter mixture.
With the mixer on a low speed, beat the flour mixture into the butter mixture about ½ at a time. The dough will be thick. If you squeeze the dough between your fingers and it sticks to them, mix in 1 more tablespoon of flour.
Mix in the chopped Oreos and chocolate chips. Ensure that the Oreos are chopped into at least quarters or fifths. If you have Oreo pieces that are too large, the cookies won't bake into nicely formed round cookies.
Form the dough into balls with about ¼ cup of dough each. I weigh my cookies for the most accurate results. Each cookie weighed about 3½ ounces or 100 grams.
Place the dough balls in a freezer bag. Freeze for at least 2 hours or up to 2 months.
When ready to bake, preheat the oven to 375℉ (190℃). Line cookie sheets with parchment or baking mats. Place the cookie dough balls (still frozen) at least 2½ inches (6 cm) apart on the lined cookie sheets.
Bake 1 sheet at a time in the middle of the preheated oven. The cookies will bake for about 13-15 minutes, or until the tops look set and just starting to turn golden. If the cookies aren't spreading enough for your liking, you can press them down slightly before baking.
Cool the cookies on the cookie tray. For perfectly round cookies, when the cookies come out of the oven take a large, round biscuit cutter or a large cup or mug. Place it around the outside of the cookie (still warm on the cookie tray). Then gently make a circular motion with the biscuit cutter around the cookie. Note - you are not cutting the cookie into a round shape - simply using the shape of the biscuit cutter/cup to mold the edges of the warm cookie into a circular shape.