Place 170 g unsalted butter in a small saucepan over medium heat.
Let it melt completely, then continue cooking. The butter will foam and bubble as the water cooks off. Swirl the pan occasionally and watch the milk solids at the bottom.
Once they turn golden brown and the butter smells nutty and toasted, remove the pan from the heat. Set it aside to cool.
While the butter cools, wash and dry 750 grams Italian prune plums. Cut each plum in half and remove the pit.
In a large bowl, whisk together: 3 large eggs, 110 g coconut sugar, 200 g full-fat Greek yogurt, 1 teaspoon vanilla extract, Zest of 1 orange, and 15 g fresh ginger. Whisk until smooth and well combined.
Slowly pour the cooled brown butter into the bowl while whisking. Keep whisking until the butter is incorporated and the mixture looks smooth.
Add 220 g all-purpose flour, 60 g light rye flour, 2 teaspoons baking powder, 11 teaspoon ground cinnamon, ½ teaspoon ground cardamom, ¼ teaspoon freshly grated nutmeg, ½ teaspoon fine salt to the bowl with the wet ingredients. Using a rubber spatula, gently fold everything together until just combined.
Don't keep mixing once you no longer see dry flour, stop. The batter should be thick and soft.
Add the halved, pitted plums directly to the bowl. Gently fold them into the batter just once or twice, being careful not to break the plums apart.
Line the bottom of the pan or springform cake pan with parchment paper and lightly grease the sides. Transfer the batter. Using your spatula, gently spread the cake batter into an even layer.
Sprinkle 1½ –2 tablespoons coconut sugar evenly over the top. The sugar will create a lightly crisp, caramelized finish as the cake bakes.
Bake at 350°F (175°C) for approximately 45–55 minutes. Start checking around 45 minutes.
The cake is done when: the top is golden brown, the center looks set rather than jiggly, and a toothpick comes out clean or with a few moist crumbs.
Let the cake cool in the pan for 10 minutes.
Serve warm or at room temperature.
Notes
Use firm, ripe plums. They should cut easily in half without falling apart. Don't overmix once you add the flour. Fold just until no streaks of flour remain.