Preheat the oven to 350ºF (175ºC) and position an oven rack to the center position.
Grease an 8x8-inch baking pan well with butter. You can use a 9x9-inch pan, but the mochi will be thinner.
In a medium bowl, whisk together the sweet rice flour, ginger, cinnamon, cardamom, cayenne, salt, and baking powder. Set aside.
Make Persimmon Purée: You’ll need 1/2 cup (140g) of persimmon purée for this recipe (I usually weigh this). You should be able to get this from 1 large or 2 medium-sized ripe hachiya persimmons. Remove the calyx from the persimmon and then rip open the persimmon. Scoop out the persimmon flesh. You can blend the persimmons in a food processor, or transfer the flesh into a jar and use an immersion blender to blend. Measure out 1/2 cup of persimmon purée.
In a large bowl, whisk together the eggs, persimmon purée, coconut milk, butter, and the brown and white sugars. Next, and the molasses and vanilla extract and whisk until smooth.
Add the flour mixture to the wet ingredients all at once. Use a wooden spoon to mix everything together, until well combined and you can only see tiny lumps of flour.
Pour the batter into the greased baking pan. Use an offset spatula or a rubber/silicone spatula to spread the batter to an even layer.
Bake the mochi for 60 to 65 minutes, rotating the pan after 30 minutes of baking. When you tap the top of the mochi with your fingernails, it should feel firm. If you're using a 9x9-inch pan, reduce the cooking time by 5 to 8 minutes) Let the mochi cool in the pan for 15 to 20 minutes before unmolding. To unmold the mochi, I like to flip the entire pan onto a wire cooling rack. Once the mochi drops from the pan, place another wire rack over the bottom of the mochi. Flip everything over once again so that the mochi is right-side up. Let the mochi cool completely before serving.
Slice the mochi into 16 or 20 pieces. Sprinkle powdered sugar on top of the mochi and serve! See notes for storing and reheating directions.