Separate the egg yolks and the egg whites by gently cracking and opening each egg, and moving the yolk back and forth between the two halves of the egg shells until all of the egg white has dripped into a cup or bowl below. You will need 6 yolks and 7 egg whites.
In a stand mixer, combine the flour, sugar, baking powder, and orange zest.
Make a well in the middle of this mixture, and add in the 6 egg yolks, vegetable oil, orange juice, and vanilla extract. Mix on medium until combined.
In a separate bowl, whisk the 7 egg whites until they are thick, white, and fluffy, with stiff peaks that have slightly droopy tips.
Gently fold the egg whites into the cake batter in 3 stages using a whisk.
Combine thoroughly, making sure there aren't any white streaks in the batter.
Pour the batter into an ungreased 10-inch chiffon or angel cake pan.
Bake for 1 hour.
When a toothpick is inserted and comes out clean, remove the pan form the oven.
Immediately turn the pan over onto a cooling rack with the cake still in it.
Allow it to cool completely, then use a knife around the edges of the pan to free the cake.
Serve with fresh orange slices and whipped cream, if desired.