Preheat oven to 350ºF. Grease and flour a 9x2" round cake pan, and line the bottom with parchment paper; set aside.
In a large bowl, whisk together flour, sugar, baking soda, baking powder, crushed cardamom, and salt.
In a separate bowl, whisk together melted butter, sour cream, milk, egg and vanilla extract. Scrape this mixture into the dry ingredients and stir until barely combined, with some floury patches remaining. Fold in 3/4 of the plums (by this time the floury patches will be incorporated - it's just important not to over-mix).
Transfer the batter to prepared cake pan, and scatter remaining plums on top. Bake for 35-45 minutes, until top feels springy when lightly pressed and a toothpick inserted in the centre comes out clean or with just a few moist crumbs. Cool for 10 minutes on a wire rack, then run a knife around the outside and invert onto a plate or clean oven mitt. Carefully then flip it right-side-up directly onto the cooling rack to finish cooling completely before transferring to a cake stand or serving plate.