Preheat oven to 350F. Butter an 8-inch cake pan and set aside.
In a large bowl combine flour baking soda, baking powder, salt, cinnamon and cardamom. Set aside. In a stand mixer fitted with a paddle attachment cream butter, sugar, molasses and vanilla.
In a blender pulse eggs and heavy cream until a pale yellow mixture forms. Add diced persimmon and pulse until a puree forms. Add puree to butter and beat until thoroughly incorporated.
Add dry mixture to wet, beating at a low speed until fully incorporated.
In a saucepan on medium heat combine butter, sugar and rum. Once mixture begins to bubble remove from heat. Pour mixture into cake pan, tilting it to ensure that the bottom is full covered. Layer persimmons in a round patter on top of rum mixture, then top with cake batter. Spread batter evenly.
Transfer cake to oven and bake for 40 minutes, until a toothpick inserted in the middle emerges clean. Allow cake to cool in pan for 15-20 minutes, or until cool enough to handle.
Loosen cake from the sides of the pan with a knife, then carefully turn over onto cooling rack. Allow to cool for an additional 15-20 minutes, then transfer to serving platter. Dust with powdered sugar and serve.