Separate 4 eggs into yolks and whites. Place the egg yolks in a bowl, and set aside on the countertop. Keep the egg whites in another large mixing bowl in the fridge.
In a separate large mixing bowl, mix cake flour and vegetable oil until smooth. Add milk and vanilla extract and mix until well combined.
Pour the egg yolks into the cake flour vegetable oil mixture. Mix until smooth. Set aside.
Line a 10”x15” jelly roll pan with parchment paper. Preheat oven to 340°F (170°C).
Take the egg white mixing bowl from the fridge, and add cream of tartar and one-third of the sugar into the egg whites. Use a hand mixer or a stand mixer to whip the egg whites for 2 minutes.
Add half of the remaining sugar into the egg whites, and use medium-low speed to whip the egg whites for 2 more minutes until the egg whites look smooth.
Add the remaining sugar into the egg whites, and use medium-low speed to whip the egg whites until medium peak forms.
Scoop ⅓ of the egg whites into the egg yolk mixture, and fold until well combined.
Transfer the egg yolk mixture to the egg white bowl, and gently fold until the color looks even.
Pour the batter into the prepared jelly roll pan, and gently spread the batter with a spatula.
Bake at 340°F (175°C) for 10-15 minutes.
Remove the pan from the oven, and transfer the sheet cake to a wire cooling rack. Allow it to cool completely.
Once cooled, cut the sheet cake into large circles (2.25 inches each) and small circles (1.25 inches each) with cookie cutters.
Brush simple syrup onto the circle cakes and cover them with plastic wrap to avoid drying out. Set the cakes aside.