Preheat oven to 350°F. Line four half-sheet pans with parchment paper and set aside until ready to use.
Place the all-purpose flour, cake flour, baking soda, table salt, and cocoa powder in a medium bowl. Use a whisk to combine all the dry ingredients. Set the bowl aside until ready to use.
Place the unsalted butter in a large microwave-proof bowl. Heat in the microwave in 30-second intervals until the butter has melted, about a minute. Add the dark brown sugar and white granulated sugar, then use a whisk to combine. Next, add the vanilla extract and two large eggs, whisking to combine. Add the white vinegar and red food coloring. Whisk until the mixture is well emulsified.
Add the dry ingredients, about a cup at a time, and use a silicone spatula to stir them into the wet ingredients. Next, stir in the semi-sweet and white chocolate chips.
Use a two tablespoon, or one ounce, ice cream scoop to portion the dough. Place eight dough balls, about three inches apart, on each of the prepared baking pans3, the last pan will only have four to five balls of dough. Bake the pans one at a time in a 350°F oven for 12 to 14 minutes.
Remove the pan from the oven and place it on a wire cooling rack. Continue baking the rest of the pans one at a time. Allow the cookies to cool for about five minutes on the pans, then transfer them to a wire cooling rack. Store in an airtight container for up to five days.
Yield 28 cookies