Whisk together the cake flour, bread flour, baking soda, baking powder, and salt until evenly combined. The mixture should look uniform with no streaks.
Beat the butter, brown sugar, and granulated sugar for 4–5 minutes until pale, fluffy, and creamy. This step creates air for a lighter cookie.
Mix in the eggs one at a time, then add the vanilla extract. The mixture should look smooth and silky.
Add the dry ingredients on low speed and mix only until no dry flour remains. Fold in the chocolate gently with a spatula.
Scoop the dough into large portions, about 3½ ounces each, and place them on a tray. Cover and refrigerate for at least 24 hours and up to 72 hours.
Preheat the oven to 350°F (175°C). Arrange six chilled dough balls on a parchment-lined baking sheet, leaving plenty of space between them.
Bake for 18–20 minutes until the edges are deep golden brown while the centers still look slightly soft. Do not overbake.
Let the cookies cool on the baking sheet for 10–15 minutes before transferring them to a cooling rack. They will finish setting while staying chewy inside.