Place an oven rack in the top third of the oven and heat to 350ºF. Cut a piece of parchment paper into 2 (9-inch) rounds. Butter the bottom and sides of 2 (9-inch) round aluminum cake pans, line with the parchment circles, lightly flour the sides, and set aside.
Sift together the cake flour and baking powder onto a paper plate or flexible cutting board. Set aside. Beat the butter in a stand mixer fitted with the paddle attachment on low speed for 1 minute. Add the sugar and salt, increase the speed to medium, and beat until the mixture is light and fluffy, about 4 minutes. Stop the mixer and scrape down the sides of the mixing bowl.
Return the mixer to medium speed and slowly add the egg yolks. Mix until completely incorporated, about 2 minutes. Stop the mixer again and scrape down the sides of the bowl. With the mixer on low speed, slowly add half of the flour mixture. Add half of the milk and the vanilla, stopping to scrape down the sides of the bowl occasionally. Add the remaining flour, followed by the remaining milk, again stopping to scrape the sides of the bowl and making sure everything is fully incorporated.
Divide the mixture evenly between the two prepared pans. Bake until the cakes reach an internal temperature of 210ºF or a toothpick inserted into the center comes out clean, 25 to 30 minutes. Remove from the oven and cool the cakes on a wire rack for 15 minutes. De-pan and cool completely, about 45 minutes, before frosting.