In a medium bowl whisk together the flour, cake flour, baking powder, baking soda and salt. Be sure to sift the cake flour first - as it is lumpy. Set aside.
In a separate large bowl, beat together the butter, brown sugar and granulated sugar until creamed (about 1-2 minutes).
Beat the eggs and vanilla extract into the butter mixture.
Add about 1/2 of the flour mixture into the butter mixture. Do a few stirs by hand, then beat together with the electric mixer on a low speed. Repeat the process with the rest of the flour mixture. It should be somewhat difficult to incorporate all of the flour. If it's impossible, then the flour wasn't measured correctly.
Fold in the chocolate chips. The batter will be thick.
Cover the bowl with plastic and refrigerate for at least 1 hour or up to 48 hours.
When ready to bake, preheat the oven to 375F (190C) and line cookie sheets with parchment paper or baking mats.
Form the dough into balls with about 1/4 to 1/3 cup of dough - I made each of mine weigh 4 ounces or 112 grams. If the dough has been in the fridge overnight, you will likely need to let it sit on the counter for about 10-15 minutes before forming into balls so that it's easier to work with. Place the dough balls at least 2 inches (about 5 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 14-16 minutes, or until the tops look set and are starting to golden. Watch carefully. Note that bake time will vary depending on your oven and how gooey you like your cookies.
Cool the cookies on the cookie sheets. Because the cookies are so thick, you can't immediately transfer them to a cooling rack or they can fall apart.