In a food processor, place the plums, lemon juice and vanilla and blend until smooth. Set aside.  

Place the sugar and water into a medium saucepan.  Cook over medium high heat until the sugar dissolves and the mixture bubbles, about 5 minutes.  Add the plum puree slowly and whisk to combine. 

Cook on medium-low about 5 minutes more until thickened (it will thicken as it cools, too).  Set aside to cool.  Once cool, refrigerate, covered, for up to 1 week.

Preheat oven to 350 degrees F.  Oil a 9-inch springform pan, line it with parchment paper, and place it on a baking sheet to catch drips.

Stir together the flours, baking powder and soda, cardamom, cinnamon, and coconut sugar.  Mix in the olive oil and egg yolks, followed by the coconut milk.  Stir until a relatively lump-free batter forms. If the batter is very thick, add another tablespoonful of coconut milk.

Spread half the batter into the prepared pan.  Top with ⅓ cup of plum sauce. Top the sauce with half of the plum wedges arranged evenly over the surface.

Pour the remaining batter on top, spread over the plums, then arrange the rest of the plums on top. Sprinkle with the reserved tablespoon of sugar.

Bake for an hour to 1 hour 20 minutes.  The cake is done when the edges are golden brown and a toothpick inserted in the center comes out clean (though it’s tricky to get a clean toothpick since the caramel sauce in the center tends to coat it).

Serve warm or allow to cool completely in the pan.  Store the cake in an airtight container at room temperature for up to 5 days.  Be sure to serve it with extra sauce.