Cut a piece of parchment paper into a 9” round, trimming it to make sure it fits perfectly in the bottom of the pan.
Grease the bottom of the springform with 1 t-spoon of the olive oil and set the parchment paper inside. Spread the remaining 2 t-spoons of olive oil on top of the parchment and sprinkle over the turbinado sugar.
Halve and pit the plums (if the pit is difficult to remove, cut the plum half with the pit in half again and pop it out). Slice the plums into thin slices—about 1/8” thick. Decoratively fan them around the bottom of the pan. Stick the pan in the freezer for about 15 – 20 minutes to help set them—this makes it easier to spread the batter without moving the plums too much.
Preheat the oven to 350°.
Pulse the dry ingredients—almond flour, coconut flour, sugar, baking powder, cinnamon, and salt—in a food processor a few times to combine. Mix the eggs, olive oil, lemon zest, and vanilla in a liquid measuring cup and whisk just to combine.
Turn the processor on and immediately pour in the wet ingredients. Let it run for just a few seconds, then scrape down the bottom and sides with a rubber spatula and pulse a few more times to make sure it’s combined.
Tip the batter into the springform and spread it evenly over the plums. Set the pan in a baking sheet and bake for about 40 – 45 minutes, until cooked through.
Remove the pan from the oven and place on a wire rack to cool for about 15 minutes. Run a sharp pairing knife around the edge of the cake to make sure it’s loosened from the pan before removing the sides. Let it cool completely—at least an hour—before inverting on to a plate, carefully removing the bottom of the pan and peeling off the parchment paper to reveal the plums.
I like to serve this just as is, but there’s nothing stopping you from keeping a bowl of whipped cream or crème fraîche (maybe lightly sweetened with powdered sugar) close by.
Make sure the baking powder is gluten free. Most use cornstarch for anti-caking, but some use wheat-based starches so err on the side of caution.
Don’t let the processor run for too long, especially once the wet ingredients are added. You run the risk of almost turning the almond flour into almond butter.
You also don’t need to use the food processor for this, just simply whisk the dry ingredients together and then the wet in a separate bowl until combined (taking care not to beat too much air into it). Pour wet into dry and stir together with a rubber spatula.
You can make this in advance! Once unmolded, flipped, and cooled completely cover it with plastic wrap and refrigerate. It's best to serve within 1 - 2 days, taking it out of the fridge and letting it come to room temp for about 30 minutes beforehand. After a couple days the plums start to dampen the top of the cake—it's not a bad thing though, and I'm still happy to eat it this way myself, even for breakfast.