In a large bowl, sift together the flour, cake flour, cocoa, baking soda, baking powder and salt. Give the bowl a whisk and set aside. Be sure to measure the flour and cake flour properly - either using a kitchen scale or whisk first, then spoon into dry measuring cups and level off the top.
In a separate large bowl, beat together the butter, brown sugar and granulated sugar until creamed.
Mix the eggs, vanilla, vinegar and food coloring into the butter mixture. It will be very vivid at this stage.
With the mixer on a low speed, beat the dry ingredients into the butter mixture. I typically do this about ½ at a time. The dough will be very thick. You may need to use your hands to get all the dry ingredients incorporated.
Stir in the white white chocolate chips.
Cover the bowl and place in the fridge for at least 2 hours or up to 48 hours.
When ready to bake, preheat the oven to 375℉ (190℃). Line cookie sheets with parchment paper. If the dough has been in the fridge overnight, you may need to let it sit on the counter for 5-10 minutes first so that it's easier to work with.
For the dough into large balls - you can use a ¼ cup or ⅓ cup measuring up. I like to weigh mine - each was 3.5 ounces (100 grams). Place 2½ inches (about 6 cm) apart on the lined cookie sheets.
Bake 1 sheet at a time in the middle of the preheated oven. Keep the other cookie dough balls in the fridge as you bake 1 sheet at a time. The cookies will take about 12-15 minutes, or until the tops look set.
Cool the cookies on the cookie sheets for at least 15 minutes. Note: because these cookies are very thick, do not try to transfer them to a cooling rack quickly or they will fall apart. For perfectly round cookies, when the cookies are fresh from the oven take a large biscuit cutter or very large mug (big enough to fit around the cookie without slicing it) and place around a cookie. Then make a gentle circular motion to trace the outside of the cookie.