Line a baking sheet with parchment paper or a nonstick baking mat and preheat oven to 425 (or according to package directions). Gently roll out pastry onto the lined baking sheet.
Thinly slice persimmons into slices about 1/2" thick. Spread pudding in the center of the pastry dough, then layer persimmons, overlapping slightly, to create a circle in the middle of the dough (see video for reference).
Gently fold the edges of the pastry dough inwards, creating small pleats where the dough overlaps. Continue all the way around the circle, then brush the pastry dough crust with melted butter, sprinkle with sugar and bake in the oven for 10-12 minutes, or until the pastry is golden brown and the persimmons are tender.
To finish, brush the persimmon filling and the crust with a final glaze of simple syrup.