Preheat the oven to 350 degrees F.
Grease the inside of an 8-inch springform pan with butter.
Place parchment paper on the bottom and butter the edges. You can use a 9-inch cake pan, but I find that it makes rather flat cake.
In a medium bowl, mix together the butter and 1 cup of sugar until the mixture becomes light and fluffy (use an electric hand mixer, stand mixer or food processor).
You can use an electric mixer, or achieve the same result with a wooden spoon and some manual effort, which takes around 3 minutes.
Add the eggs one by one, making sure they are fully integrated after each addition.
Carefully sift in the flour, baking powder, and salt, then mix until the ingredients are just blended.
Pour the batter into the pan and smooth the surface evenly.
Arrange the halved plums with their cut sides facing up, and gently press them into the batter using your fingers.
Mix together the topping ingredients: sugar, cardamom, and cinnamon, then sprinkle the mixture evenly over the top of the cake.
Put the cake in the oven and bake for around 50 minutes to an hour, or until it becomes a golden brown color and a toothpick or cake tester inserted into the center comes out clean.
Allow the cake to cool on a wire rack for a minimum of 30 minutes before cutting.
Enjoy it either warm or at room temperature. Alternatively, you can refrigerate the cake overnight for a different experience; serving it cold is equally enjoyable.