Make the prune puree. Add the prunes and hot water to a high-powered blender and blend until smooth. Measure out 1/2 cup and refrigerate the rest for up to 4 months for use in a variety of recipes. Clean the blender carafe.
Make the persimmon puree. If very ripe and soft there is no need to peel the Fuyu persimmons. Simply remove the leafy top and toss the persimmon into the blender. If it is still very firm peel each Fuyu, trim off the leafy top and chop roughly. Add the pieces to the high-powered blender and blend until smooth, 1 to 2 minutes. Measure out 2 cups and save any extra persimmon puree for another use.
Heat the oven to 350°F (180℃) with a rack in the middle.
Mix the dry ingredients. Whisk the flour, baking powder, salt, baking soda, cinnamon, ginger, nutmeg, allspice, and orange zest in a large bowl to break up any lumps. Butter a 13-by-9-inch (23 × 33 cm) baking dish and set it aside.
Make the batter. In another large bowl, whisk the persimmon puree, prune puree, half-and-half, brown sugar, and eggs until evenly blended. Whisk in the melted butter, dessert wine, and vanilla bean paste until just combined. Stir in the flour mixture in four additions, letting each one disappear before adding the next and scraping down the bowl as you go.
Bake. Pour the batter into the prepared dish and bake until a cake tester in the center comes out clean, 40 to 45 minutes.
Serve. Serve warm or at room temperature. Top each square with crushed pistachios, a small dollop of whipped cream, and a thin slice of Fuyu persimmon.