Preheat the oven to 180°C (350°F). Grease and flour a 22 cm (9 inch) round cake pan.
In a large bowl, whisk together the eggs and granulated sugar until the mixture is pale and fluffy.
Add the yogurt, vegetable oil, and vanilla extract to the egg mixture and whisk until combined.
In a separate bowl, sift together the flour, baking powder, and salt.
Gradually add the dry ingredients to the wet ingredients, folding gently with a spatula until just combined. Do not overmix.
Pour half of the batter into the prepared cake pan and spread evenly.
Arrange half of the sliced plums over the batter in a single layer.
Pour the remaining batter over the plums and spread gently to cover.
Top with the remaining plum slices, pressing them lightly into the batter.
Bake in the preheated oven for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and let it cool in the pan for 10 minutes.
Transfer the cake to a wire rack to cool completely.
Once cooled, dust the top with powdered sugar before serving.