Preheat the oven to 350°F and move rack to upper third. Grease and line one 9×13″ rectangular cake pan.
In a large bowl, sift together the cake flour, baking powder, and salt.
In the bowl of a stand mixer fitted with the paddle attachment, cream together the butter and sugar until light and fluffy on medium-high. This usually takes about 5 minutes. Scrape down the sides and bottom.
With the mixer on medium-low, add the egg yolks one at a time, followed by the vanilla bean paste. Scrape down the sides and bottom again.
Turn the mixer to low and add the flour mixture in three parts, alternating with the milk. Begin and end with the flour mixture.
When the last of the flour has been incorporated, scrape down the sides and bottom of the mixer bowl one more time. Beat on medium for about 20 seconds to fully mix everything.
Divide the batter between the two pans and bake for 30-35 minutes, or until a cake tester comes out clean in the center.
Cool the cakes in their pans on a rack for 10 minutes before flipping them out and letting it cool completely on the rack.
If you’re making the cakes ahead of time, wrap them in cling film after they’ve cooled and store in an airtight Ziploc bag in the fridge until assembly time.