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 becomes pale and fluffy.
Add the plain 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, 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 evenly.
Top the cake with the remaining plum slices arranged evenly on the surface.
Sprinkle 2 tablespoons of granulated sugar evenly over the top to create a sweet crust.
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 allow it to cool in the pan for 10 minutes before transferring it to a wire rack to cool completely.
Serve the plum cake at room temperature, optionally dusted with powdered sugar.