Preheat your oven to 350°F.
In a large mixing bowl, whisk together the oil, yogurt, sugar, eggs, and vanilla.
To the bowl, add the cardamom, flour, baking powder, and salt. Gently mix or fold together until just combined. It’s okay if there are some smaller lumps.
Pour the cake batter into a prepared baking pan (lined with parchment paper, sprayed or buttered as needed). Use a rubber spatula to spread the batter into an even layer. Place the sliced plums in a circular pattern on top of the cake.
Pop the cake into a preheated oven and bake for 50 minutes until golden brown and a toothpick comes out dry with only a few crumbles.