Preheat the oven to 350°F. Line the bottom of a springform pan with parchment, then butter and flour the sides. Set aside.
In a medium bowl, whisk together the flour, baking powder, cardamom, and salt. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, cream together the butter and the sugar for 2 to 3 minutes until it is light and fluffy. Scrape down the sides and the bottom of the bowl with a spatula as needed.
Add the eggs, one at a time, beating well between each addition. Add the vanilla and mix well. Beat the mixture for 2 minutes until it is aerated and very light in colour. Scrape down the sides of the bowl and make sure the batter is evenly mixed at this point. You can beat it a little more to make sure.
Stir in the flour mixture on low speed, just until the flour is completely incorporated.
Spread the batter in the prepared pan and smooth the top. Place the plum halves, cut side down (or up works too!), and pressing them slightly into batter. Scatter a few sliced almonds and some turbinado sugar on top (if using)
Bake the cake until the edges are golden brown and the plums are soft and releasing their juices. A cake tester comes out clean when inserted in the cake. This could take as much as 90 minutes, especially if you placed the plums cut side down on the cake batter.
Remove the pan from the oven, and let cool on a wire rack for 10 minutes, then release the cake from the springform pan.
Let the cake cool completely before slicing and serving.