Whisk together the all-purpose flour, cake flour, baking powder, and baking soda in a bowl. Set aside.
Beat the butter, granulated sugar, and brown sugar in a stand mixer on medium speed until pale, fluffy, and noticeably increased in volume, about 3 to 5 minutes.
Add the eggs one at a time, beating after each addition. Beat in the vanilla.
Add the flour mixture in 5 to 7 additions on low speed, mixing until just combined. Stir in the chocolate chips.
Chill the dough for at least 2 hours. Portion into balls approximately the size of half a lemon for full-size cookies, or smaller if baking from frozen.
Preheat the oven to 375°F. Bake for 13 to 14 minutes, until the edges are set and golden and the centers still look soft and slightly underdone.
Cool on the sheet pan for 10 minutes before transferring to a wire rack. The centers will firm up as they cool.
Notes
Doneness cue: Pull the cookies when the edges are golden but the centers still look almost raw. They will set fully on the hot pan during the 10-minute rest. Chilling: Chilling the dough for at least 2 hours produces taller, thicker cookies. The cold fat melts more slowly, giving the cookies time to set before they spread. Mixer: Use a stand mixer. This dough is thick enough to strain a hand mixer motor. If using a hand mixer, stop adding flour the moment it is fully incorporated. Storage: Store baked cookies in an airtight container at room temperature for up to 5 days. Freeze baked cookies in a single layer, then transfer to a zip-close bag for up to 2 months.