Preheat the oven to 350°F [180°C]. Butter a 9 in [23 cm] round cake pan. Cover the bottom with a parchment round.
Prep the fruit: Combine the olive oil, brown sugar, and cinnamon in a small mixing bowl. Stir together until completely combined. Pour into the prepared cake pan. Use an offset spatula to spread everything out evenly until the bottom is covered. Arrange the plums in a single layer over the buttery brown sugar. Set aside.
In a medium mixing bowl, sift together the flour, baking powder, baking soda, salt, and cinnamon.
In a large mixing bowl, beat the eggs and olive oil together until very well incorporated. Then add the brown sugar, milk, and yogurt, and beat to combine well.
Pour the wet mixture over the dry mixture and stir together just until everything is combined. It won’t be completely smooth, and there might be some tiny lumps; this is completely fine. Don’t overmix!
Carefully pour the batter over the plums, smooth out the top a little bit with a spatula, and bake for about 35 minutes, until a toothpick inserted into the center comes out clean.
Let cool for 5 to 10 minutes in the cake pan. Then trace around the edge with a butter knife to make sure it’s loosened from the pan. Invert onto a serving plate, let cool for at least 30 minutes more, and serve.
* In A Dish for All Seasons, I use buttermilk, which you can use here in place of the yogurt and milk (just use 1/2 cup [120 g] buttermilk in place of both).