Preheat your oven to 180°C (350°F). Grease your cake pan and line the bottom with parchment paper to prevent sticking.
In a large mixing bowl, whisk together the homemade yogurt and sugar until smooth and slightly frothy—this should take about 2 minutes and fill the bowl with a sweet, tangy aroma.
Add the oil to the mixture and whisk until fully incorporated, creating a smooth, glossy base.
One at a time, beat in the eggs, whisking well after each addition. The mixture will become more voluminous and glossy, indicating good emulsion.
Sift together the flour, baking powder, and vanilla extract into a separate bowl. Gently fold the dry ingredients into the wet batter using a spatula, taking care not to overmix—just until no streaks of flour remain.
Pour the batter into your prepared cake pan, smoothing the top with the spatula. Lightly tap the pan on the counter to release any trapped air bubbles.
Bake for 30 to 35 minutes, or until the cake is golden on top and a toothpick inserted into the center comes out clean. Keep an eye on it after 30 minutes to prevent overbaking.
Once baked, remove the cake from the oven and let it cool in the pan for 10 minutes. Then, transfer it to a wire rack and cool completely for about an hour, allowing the crumb to set and develop its moist texture.
When cooled, slice the cake with a serrated knife to reveal a tender, moist crumb with a lively, tangy aroma. Serve plain or with a dusting of powdered sugar for an elegant finish.
