Preheat the oven to 180°C/160°C Fan/350°F. Grease the base and sides of a 18 cm (7-inch) round loose-based or springform tin with butter and line the base with non-stick baking paper (if using a loose-based tin, you may wish to line the sides too).
First make the lemon sugar: Add the sugar and lemon zest to a large bowl. Rub the two together with your fingertips until the sugar is moist, lightly lemon coloured and filled with the zesty lemon fragrance. Set aside.
Next, prepare the plums: slice them in half and remove the stone. Cut into slices (not too thin, not too thick). I like a chunkier slice but you can slice it to your preference.
Add the eggs, vanilla extract and almond extract to the lemon sugar and whisk until pale, light and bubbly - this takes approximately one minute. You can use an electric whisk if you like but definitely not necessary for this cake.
Add the yogurt, followed by the olive oil, and whisk until fully combined.
Sift in the flour, baking powder, and salt and, using a spatula, fold gently until a few streaks of flour remain. Small lumps in the batter are totally fine - don't be tempted to over-mix!
Pour the batter into the prepared tin and smooth over the surface using a spatula. Arrange the plum slices on top, on their sides. Sprinkle it with demerara sugar (this adds a delicious crunch).
Bake in the oven for about 50-55 minutes - or until golden brown and the sponge springs back or a toothpick inserted into the centre comes out clean. Have a look at about 45 minutes - if the cake is browning too quickly, then cover in a tent of aluminium foil and continue to bake.
Remove from the oven and leave to cool on a wire rack for 10 minutes. Then carefully run a butter knife around the edge and remove the cake from the tin. Return to the wire rack to finish cooling completely.