In a large bowl, sift together the flour, cake flour, baking powder, baking soda and salt. Whisk together and set aside.
In a separate large bowl, beat together the butter, brown sugar and granulated sugar until creamed.
Beat the eggs and vanilla extract into the butter mixture.
Gradually mix the dry ingredients into the butter mixture starting with the mixer on a low speed. The dough will be thick. If you're not able to incorporate all of the flour, it's likely that you overmeasure the flour. If that's the case, mix in 1-3 teaspoons (not tablespoons) of milk.
Using a rubber spatula, stir the freeze dried berries and white chocolate chips into the dough.
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 or silicone baking mats.
Form the dough into balls with about 1/4 to 1/3 cup of dough each. I use a scale to weigh my cookie dough balls - each weighed 3.5 ounces (100 grams). Note that if the dough was in the fridge overnight, you may need to let it sit on the counter for 10 minutes first so that it's easier to work with.
Place the cookies 3 inches (about 7-8 cm) apart on the cookie sheets. Bake 1 sheet at a time in the middle rack of the oven. They'll need about 13-15. Remove from the oven when the tops look set and slightly golden.
Cool the cookies on the cookie tray.