In a blender or food processor, pulse chopped persimmons until smooth. Measure ¾ cup and save or discard any remaining pulp.
In a large bowl, mix flour, baking soda, ½ teaspoon cinnamon, nutmeg, salt, allspice and cloves.
In a separate bowl, use a handheld mixer or stand mixer to cream butter and sugar until light and fluffy. Add starter, persimmon pulp and eggs. Beat on medium speed until combined.
Add dry mixture and combine until fully incorporated.
Line a cupcake pan with paper or silicone baking cups and fill until ¾ full. Bake for 20-25 minutes or until a toothpick inserted in the center of the cupcake comes out clean. Remove from pan and allow to cool completely before frosting.
Sprinkle frosted cupcakes with cinnamon and top with candied ginger.