Preheat the oven to 350 degrees F. Grease and flour three 8-inch cake pans and line the bottoms with parchment paper.
Sift together the cake flour, baking powder, and salt into a large bowl.
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter on medium speed for 2 minutes. Add the sugar and mix on medium-high until light and fluffy, 3 to 5 minutes. Stop the mixer and scrape down the bowl.
Turn the mixer to medium-low and add the vanilla. Add the egg yolks and eggs one at a time, mixing until each is incorporated before adding the next. Mix until combined. Stop the mixer and scrape down the bowl.
Turn the mixer to low and add the flour mixture in three batches, alternating with the milk, beginning and ending with the flour mixture. After the last streaks of the flour mixture are combined, mix on medium for no more than 30 seconds. Stop the mixer and fold in the chocolate by hand.
Evenly divide the batter among the three prepared pans. Bake for 28 to 35 minutes, until a toothpick inserted into the center of each cake comes out clean. Let the cakes cool on a wire rack for 10 to 15 minutes before removing from the pans. Allow the cakes to cool completely, right-side up, on the wire rack before removing the parchment.