Step 1: Preheat your oven to 350°F (175°C). Grease and flour a 9-inch springform pan, or line the bottom with parchment paper.
Step 2: In a medium bowl, cream together the softened unsalted butter and 3/4 cup granulated sugar until light and fluffy.
Step 3: Beat in 1 large egg and vanilla extract until well combined.
Step 4: In a separate small bowl, whisk together the all-purpose flour, baking powder, ground cinnamon, ground cardamom, and 1/4 teaspoon salt.
Step 5: Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Do not overmix.
Step 6: Spread the cake batter evenly into the prepared springform pan.
Step 7: Arrange the sliced plums decoratively over the cake batter.
Step 8: In another bowl, whisk together the milk, heavy cream, 2 large eggs, 1/4 cup granulated sugar, and a pinch of salt until smooth.
Step 9: Carefully pour the custard mixture over the plums and cake batter in the pan.
Step 10: Bake for 55-65 minutes, or until the cake is golden brown, the custard is set (a slight wobble in the center is okay), and a toothpick inserted into the cake portion comes out clean.
Step 11: Let the cake cool completely on a wire rack before removing the springform pan.
Step 12: Slice and serve at room temperature or chilled.