Preheat the oven to 350°F (180°C/160°C fan). Grease the base and sides of a 7-inch (18 cm) round springform or loose-bottom cake pan with butter and line the base with parchment paper (if using a loose-bottom tin, you may wish to line the sides too).
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-colored, 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 1 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. 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 pan 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 center comes out clean. Have a look at about 45 minutes - if the cake is browning too quickly, then cover in a tent of aluminum 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 pan. Return to the wire rack to finish cooling completely.