Using a stand mixer fitted with the paddle attachment, combine butter, light brown sugar, and granulated sugar. Beat on medium speed until the mixture is creamy and no visible butter chunks remain.

Add the egg and egg yolk. Mix until fully combined. Add the vanilla extract and continue mixing, scraping down the sides of the bowl as needed.

In a medium bowl, whisk together all-purpose flour, cake flour, kosher salt, baking powder, and baking soda.

With the mixer on low speed, gradually add the dry ingredients to the wet ingredients. Mix until mostly incorporated.

Shape the dough into 8 large balls.

Place 4 cookie dough balls onto each prepared baking sheet, spacing them out evenly.

Bake for 12 to 13 minutes, or until the edges are golden brown.

Let the cookies cool on the baking sheet for 10 minutes before transferring to a wire rack.

Serve warm.