Preheat the oven to 350 degrees Fahrenheit. Position the oven rack in the bottom third of the oven.
Separate all six eggs, placing the egg whites in a large bowl that has been wiped clean with a paper towel moistened with white vinegar. Place the egg yolks in a medium bowl and set aside.
Add the salt to the egg whites and beat at medium-high speed using a hand mixer or stand mixer until foamy. Add the cream of tartar and continue to beat at high speed, adding the ¾ cup sugar a few Tablespoons at a time until the mixture forms stiff, glossy peaks. Set the meringue aside as you prepare the egg yolk mixture.
Add the warm water to the egg yolks and mix on medium speed using a whisk or mixer. Add ½ cup sugar, vanilla extract, baking powder, and flour. Mix until the batter is completely smooth.
Using a spatula, gently move a portion of the meringue away from the side of its mixing bowl. Add the vanilla and ½ cup flour to the space created by moving the meringue. This step prevents the weight of the flour from deflating the airy meringue. Using the spatula, gently fold the meringue until the flour and vanilla extract are fully incorporated and the mixture is smooth.
Transfer two thirds of the meringue mixture to an ungreased angel food cake pan, spreading lightly if necessary to cover the bottom of the pan. Add the egg yolk mixture to the pan. There is no need to spread the yolk mixture or completely cover the meringue. Add the remaining meringue to the pan. Using a skewer or toothpick, lightly swirl the two batters by moving in a random pattern around the pan.
Transfer the pan to the preheated oven and bake for 35-40 minutes. When fully baked, a toothpick inserted into the center of the cake should come out with crumbs attached. Remove the pan from the oven and allow the cake to cool.
Once cool, run a sharp knife or small metal offset spatula around the outside of the pan to loosen the cake. Invert the cool cake onto a plate. Slice the cake into slices and serve plain or dressed with fresh berries and whipped cream.
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