Preheat the oven to 180°C (350°F). Grease and flour a 22 cm (9 inch) round cake pan.
In a large mixing bowl, whisk together the eggs and granulated sugar until the mixture is pale and slightly fluffy.
Add the yogurt, vegetable oil, and vanilla extract to the egg mixture and whisk until well combined.
In a separate bowl, sift together the all-purpose flour, baking powder, ground cinnamon, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing 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, then pour the remaining batter on top and smooth the surface.
Top with the remaining plum slices, gently pressing them 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 to a wire rack to cool completely.
Once cooled, dust the top of the cake with powdered sugar before serving.