Heat the oven to 350°F Coat 9x13 pan with butter and flour, and tap out any excess flour. Set aside.
Combine flour, baking powder, salt, and baking soda in a small bowl and whisk break up any lumps. Set aside.
Place butter in the bowl and mix on medium-high speed until fluffy and light in color, about 3 minutes. Add sugar and vanilla, and continue to beat another 5 minutes. Add yolks one at a time, letting each incorporate fully before adding the next. Scrape down the sides of the bowl.
Add whole eggs in the same manner, allowing 1 minute between additions. Scrape down the sides of the bowl and the paddle.
Add 1/2 of the flour mixture, and turn the mixer to low speed, mixing until the flour is just incorporated. Add 1/2 of the milk, and mix until just incorporated. Continue with remaining flour mixture and milk, until all ingredients are incorporated and smooth.
Pour in prepared pan and bake until cake edges slightly pull away from the pan and a cake tester inserted in the center comes out dry with just a few crumbs, about 35 to 40 minutes.
Remove the pan from the oven and let cool on a wire rack, about 10 to 15 minutes. Run a knife around the perimeter of cake, and turn out onto the rack to cool completely (at least 1 1/2 hours) before frosting.