Preheat the oven to 350 degrees Fahrenheit/ 180 degrees Celsius. Grease a springform all over, line only its bottom with parchment paper.
Cut plums: Wash and dry the plums. Depending on the sort you use, halve or quarter them. Large plums should be quartered, while Italian plums should be halved. Set them aside.
Wet ingredients: Place eggs, sugar, and oil in a large bowl. Mix for about 3 minutes until frothy. Stir in the yogurt and vanilla extract.
Batter: Sieve the flour, salt, and baking powder over the egg mixture and fold in shortly (Note 3).
Assemble: Pour the batter into the prepared form. Arrange the plums on top and press them slightly into the batter (not too much, they should not be covered with batter).
Bake for 50 to 60 minutes. Cover the cake loosely with aluminum foil during the last 15 minutes if it starts to get too dark (it depends on your oven).
Check: Insert a toothpick in the middle of the cake, between the plums. It should come out clean.
Rest for about 15 minutes in the tin. Remove and let cool on a wire rack (Note 4).
Dust with icing sugar before serving.