Heat the oven to 300°F (150°C). Place the oven rack in the center. Line a 6-inch round cake pan with parchment paper. Let the paper come a little above the sides. This makes the cake easier to lift out later.
Place the 3 egg yolks in a mixing bowl. Add the 1 cup yogurt, 1 teaspoon lemon juice, 1 teaspoon vanilla extract, and 1/8 teaspoon salt. Whisk until the mixture looks smooth. There should be no streaks left.
Sift the cake flour and cornstarch into the bowl. Stir with a rubber spatula until everything comes together. Stop mixing as soon as you no longer see dry flour. The batter should look smooth and thick.
Place the 3 egg whites in a clean, dry bowl. Beat them until they become foamy. A clean bowl helps the egg whites whip much better.
Slowly add the 1/3 cup sugar while beating the egg whites. Keep mixing until stiff peaks form. The peaks should stand up when you lift the beaters. This gives the cake its light texture.
Add about one-third of the whipped egg whites to the yolk mixture. Fold gently with a spatula. Add the rest in two more batches. Fold only until no white streaks remain. Gentle folding helps keep the batter full of air.
Pour the batter into the prepared cake pan. Smooth the top with the spatula. Use a toothpick to pop any large air bubbles you can see.
Place the cake pan inside a larger baking pan. Carefully pour hot water into the larger pan. The water should come about halfway up the sides of the cake pan. This helps the cake bake slowly and evenly.
Bake at 300°F (150°C) for 20 minutes. Without opening the oven, lower the heat to 285°F (140°C). Bake for another 40 minutes. The center should look set with a slight wobble.
For a little more color, broil the cake for 1 to 2 minutes. Watch it closely. The top can brown very quickly.
Take the cake out of the oven. Leave it in the pan for several minutes. Then move it to a cooling rack. Let it cool completely before removing the parchment paper. The texture becomes even softer as it cools.
Place the cooled cake on a serving plate. Dust the top with icing sugar just before serving. Slice with a clean knife and serve. Every piece should be soft, light, and airy.