Preheat oven to 350F and position a rack in the center.
Use a baking spray with flour to grease a 9 inch round cake pan. You can use unsalted butter and flour as a substitute.
Combine the flour and baking powder in a large bowl and set aside.
Heat the milk and butter in a small saucepan over low heat. As soon as the butter is melted remove from the heat and add the vanilla. Set aside.
Using the whisk attachment of a stand mixer, whip the eggs, egg yolks, and sugar on high speed for 3 minutes. The mixture will triple in volume and turn a very pale yellow.
Reduce the speed to medium and continue whipping for 3 minutes. The mixture will continue to increase in volume and thicken.
With the mixer on the lowest speed, slowly sprinkle ¼ cup of the flour mixture over the eggs and mix until no trace of white remains. Repeat with the remaining flour mixture in 2 additions; reserve the empty flour bowl.
When the flour is fully incorporated, the batter will be thick and stiff. If you let it pour off the whist into the bowl, it will fall back on itself like a ribbon.
Pour half of the batter into the empty flour bowl.
Pour the warm milk mixture into the batter and stir gently to combine.
Slowly pour the milk/batter mixture back into the mixing bowl, and using a rubber spatula, fold the mixtures together quickly but gently until fully blended. You will see tiny air bubbles appear on the surface; work quickly to prevent too much air from escaping.
Pour the batter into the prepared pan and place on the center rack of the oven.
Bake for 30-40 minutes, or until the edges pull back from the pan and a toothpick inserted in the center of the cake comes out clean.
Set the cake on a rack to cool.
When the cake is cool, run a knife around the edge of the pan and turn the cake out onto a plate.