Preheat the oven to 180°C (350°F). Grease and flour a 22 cm (9 inch) round cake pan.
Wash the plums, remove the pits, and cut them into thin slices. Set aside.
In a large bowl, whisk together the eggs and granulated sugar until pale and fluffy.
Add the yogurt, vegetable oil, and vanilla extract to the egg mixture and mix until combined.
In a separate bowl, sift together the flour, baking powder, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing gently until just combined. Do not overmix.
Pour half of the batter into the prepared cake pan and spread evenly.
Arrange half of the plum slices over the batter in the pan.
Pour the remaining batter over the plums and spread evenly.
Top the cake 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, then transfer it to a wire rack to cool completely.
Before serving, sprinkle the cake with powdered sugar.